To describe the constant coefficients of Ek(χ,1)Ek(χ,1)E_(k)(chi,1)E_{k}(\chi, 1)Ek(χ,1) we first set SSSSS to be minimal, i.e., the union of S∞S∞S_(oo)S_{\infty}S∞ and Sram Sram S_("ram ")S_{\text {ram }}Sram , where the latter is the set of primes dividing nnn\mathfrak{n}n. Next we assume for the remainder of the article that n≠1n≠1n!=1\mathfrak{n} \neq 1n≠1; the case n=1n=1n=1\mathfrak{n}=1n=1 causes no difficulties but the formulas must be slightly modified. We then have
Here ΘS=ΘS,ϕ(H/F,0)∈Q[G]ΘS=ΘS,Ï•(H/F,0)∈Q[G]Theta_(S)=Theta_(S,phi)(H//F,0)inQ[G]\Theta_{S}=\Theta_{S, \phi}(H / F, 0) \in \mathbf{Q}[G]ΘS=ΘS,Ï•(H/F,0)∈Q[G] denotes the SSSSS-depleted but unsmoothed Stickelberger element. We have Ek(χ,1)∈Mk(n,χ;Rp)Ek(χ,1)∈Mkn,χ;RpE_(k)(chi,1)inM_(k)(n,chi;R_(p))E_{k}(\chi, 1) \in M_{k}\left(\mathfrak{n}, \chi ; R_{p}\right)Ek(χ,1)∈Mk(n,χ;Rp) for k>1k>1k > 1k>1k>1 and E1(χ,1)∈M1(n,χ;Frac(Rp))E1(χ,1)∈M1n,χ;Fracâ¡RpE_(1)(chi,1)inM_(1)(n,chi;Frac(R_(p)))E_{1}(\chi, 1) \in M_{1}\left(\mathfrak{n}, \chi ; \operatorname{Frac}\left(R_{p}\right)\right)E1(χ,1)∈M1(n,χ;Fracâ¡(Rp)) because of the possible nonintegrality of the constant term.
6.4. Eisenstein series to cusp forms
In order to define a cusp form from the Eisenstein series, one is led to consider certain linear combinations of the analogues of Ek(χ,1)Ek(χ,1)E_(k)(chi,1)E_{k}(\chi, 1)Ek(χ,1) as HHHHH ranges over all its CM subfields containing FFFFF. This process also incorporates smoothing at the primes in TTTTT. We avoid stating the slightly complicated formula here (see [17, PRoPoSITION 8.14]), but the end result is a group ring form Wk(χ,1)Wk(χ,1)W_(k)(chi,1)W_{k}(\chi, 1)Wk(χ,1) whose constant terms are given by
where we remind the reader that Θ#=ΘΣ,Σ′#Θ#=ΘΣ,Σ′#Theta^(#)=Theta_(Sigma,Sigma^('))^(#)\Theta^{\#}=\Theta_{\Sigma, \Sigma^{\prime}}^{\#}Θ#=ΘΣ,Σ′#. Building off the computations of [19], we calculate in [17,$8][17,$8][17,$8][17, \$ 8][17,$8] the constant terms of the Wk(χ,1)Wk(χ,1)W_(k)(chi,1)W_{k}(\chi, 1)Wk(χ,1) at all cusps; the terms in (6.12) can be viewed as the constant terms "at infinity." Indeed, it is the attempt to cancel the constant terms at other cusps that leads naturally to the definition of the Wk(χ,1)Wk(χ,1)W_(k)(chi,1)W_{k}(\chi, 1)Wk(χ,1).
In order to define a cusp form, we apply two important results of Silliman [43]. The first of these generalizes a result of Hida and Wiles and is stated below.
Theorem 6.2 ([43, THEOREM 10.7]). Let mmmmm be a fixed positive integer. For positive integers k≡0(mod(p−1)pN)k≡0mod(p−1)pNk-=0(mod(p-1)p^(N))k \equiv 0\left(\bmod (p-1) p^{N}\right)k≡0(mod(p−1)pN) with NNNNN sufficiently large, there is a Hilbert modular form VkVkV_(k)V_{k}Vk of level 1, trivial nebentypus, and weight kkkkk defined over ZpZpZ_(p)\mathbf{Z}_{p}Zp such that
and such that the normalized constant term of VkVkV_(k)V_{k}Vk at every cusp is congruent to 1(modpm)1modpm1(modp^(m))1\left(\bmod p^{m}\right)1(modpm).
The idea to construct a cusp form is to fix a very large integer mmmmm and to consider the product W1(χ,1)Vk∈Mk+1(n,χ,Rp)W1(χ,1)Vk∈Mk+1n,χ,RpW_(1)(chi,1)V_(k)inM_(k+1)(n,chi,R_(p))W_{1}(\chi, 1) V_{k} \in M_{k+1}\left(\mathfrak{n}, \chi, R_{p}\right)W1(χ,1)Vk∈Mk+1(n,χ,Rp) with VkVkV_(k)V_{k}Vk as in Theorem 6.2. This series has constant terms at infinity congruent to 2−nΘ#2−nΘ#2^(-n)Theta^(#)2^{-n} \Theta^{\#}2−nΘ# modulo pmpmp^(m)p^{m}pm. One then wants to subtract off 2−nΘ#Hk+1(χ)2−nΘ#Hk+1(χ)2^(-n)Theta^(#)H_(k+1)(chi)2^{-n} \Theta^{\#} H_{k+1}(\chi)2−nΘ#Hk+1(χ) for some group ring valued form Hk+1(χ)∈Mk+1(n,χ,Rp)Hk+1(χ)∈Mk+1n,χ,RpH_(k+1)(chi)inM_(k+1)(n,chi,R_(p))H_{k+1}(\chi) \in M_{k+1}\left(\mathfrak{n}, \chi, R_{p}\right)Hk+1(χ)∈Mk+1(n,χ,Rp) to obtain a cusp form. If there exists a prime above ppppp dividing nnn\mathfrak{n}n (i.e., Σ−S∞Σ−S∞Sigma-S_(oo)\Sigma-S_{\infty}Σ−S∞ is nonempty), then this strategy works. Silliman's second result, which generalizes a result of Chai and is stated in [43, THEOREM 10.10], implies that one can obtain a form that is cuspidal at the cusps "above infinity at ppppp " in this fashion. Applying Hida's ordinary operator then yields a form that is cuspidal.
Theorem 6.3 ([17, theOREM 8.18]). Suppose gcd(n,p)≠1gcdâ¡(n,p)≠1gcd(n,p)!=1\operatorname{gcd}(\mathfrak{n}, p) \neq 1gcdâ¡(n,p)≠1. For positive integers k≡1k≡1k-=1k \equiv 1k≡1(mod(p−1)pN)mod(p−1)pN(mod(p-1)p^(N))\left(\bmod (p-1) p^{N}\right)(mod(p−1)pN) and NNNNN sufficiently large, there exists Hk(χ)∈Mk(n,χ,Rp)Hk(χ)∈Mkn,χ,RpH_(k)(chi)inM_(k)(n,chi,R_(p))H_{k}(\chi) \in M_{k}\left(\mathfrak{n}, \chi, R_{p}\right)Hk(χ)∈Mk(n,χ,Rp) such that
lies in Sk(np,R,χ)Sk(np,R,χ)S_(k)(np,R,chi)S_{k}(\mathfrak{n} p, R, \chi)Sk(np,R,χ).
The significance of Theorem 6.3 is that we have now constructed a cusp form that is congruent to an Eisenstein series modulo Θ#Θ#Theta^(#)\Theta^{\#}Θ#.
When gcd(n,p)=1gcdâ¡(n,p)=1gcd(n,p)=1\operatorname{gcd}(\mathfrak{n}, p)=1gcdâ¡(n,p)=1, the construction of the cusp form is in fact more interesting, and a new feature appears. In this case, the ordinary operator at ppppp does not annihilate the form Wk(χ,1)Wk(χ,1)W_(k)(chi,1)W_{k}(\chi, 1)Wk(χ,1), and it must be incorporated into our linear combination. Moreover, this apparent
cost has a great benefit-we obtain a congruence between a cusp form and Eisenstein series not only modulo Θ#Θ#Theta^(#)\Theta^{\#}Θ#, but modulo a multiple x⋅Θ#x⋅Θ#x*Theta^(#)x \cdot \Theta^{\#}x⋅Θ# for a certain x∈Rpx∈Rpx inR_(p)x \in R_{p}x∈Rp. This element xxxxx has an intuitive meaning-it represents the trivial zeroes of the ppppp-adic LLLLL-function associated to χχchi\chiχ, even the " modpmodpmod p\bmod pmodp trivial zeroes." The precise definition is as follows.
Lemma 6.4. Suppose gcd(n,p)=1gcdâ¡(n,p)=1gcd(n,p)=1\operatorname{gcd}(\mathfrak{n}, p)=1gcdâ¡(n,p)=1. For positive k≡1(mod(p−1)pN)k≡1mod(p−1)pNk-=1(mod(p-1)p^(N))k \equiv 1\left(\bmod (p-1) p^{N}\right)k≡1(mod(p−1)pN) with NNNNN sufficiently large, the element
lies in Sk(πp,R,χ)Sk(Ï€p,R,χ)S_(k)(pip,R,chi)S_{k}(\mathfrak{\pi} p, R, \boldsymbol{\chi})Sk(Ï€p,R,χ).
The extra factor of xxxxx in our congruence between the cusp form Fk(χ)Fk(χ)F_(k)(chi)F_{k}(\chi)Fk(χ) and a linear combination of Eisenstein series plays an extremely important role in showing that the Galois cohomology classes we construct are unramified at ppppp.
We conclude this section by interpreting the congruences of Theorems 6.3 and 6.5 in terms of Hecke algebras. We consider the Hecke algebra T~T~tilde(T)\tilde{\mathbf{T}}T~ generated over RpRpR_(p)R_{p}Rp by the operators TqTqT_(q)T_{\mathfrak{q}}Tq for primes q∤npq∤npq∤np\mathfrak{q} \nmid \mathfrak{n} pq∤np and UpUpU_(p)U_{\mathfrak{p}}Up for primes p∣pp∣pp∣p\mathfrak{p} \mid pp∣p. (We ignore the operators UqUqU_(q)U_{\mathfrak{q}}Uq for q∣n,q∤pq∣n,q∤pq∣n,q∤p\mathfrak{q} \mid \mathfrak{n}, \mathfrak{q} \nmid pq∣n,q∤p in order to avoid issues regarding nonreducedness of Hecke algebras arising from the presence of oldforms.) We denote by T=epord (T~)T=epord (T~)T=e_(p)^("ord ")( tilde(T))\mathbf{T}=e_{p}^{\text {ord }}(\tilde{\mathbf{T}})T=epord (T~) Hida's ordinary Hecke algebra associated to T~T~tilde(T)\tilde{\mathbf{T}}T~. Let ϵcyc :GF→Zp∗ϵcyc :GF→Zp∗epsilon_("cyc "):G_(F)rarrZ_(p)^(**)\epsilon_{\text {cyc }}: G_{F} \rightarrow \mathbf{Z}_{p}^{*}ϵcyc :GF→Zp∗ denote the ppppp-adic cyclotomic character of FFFFF. Theorems 6.3 and 6.5 then yield:
Theorem 6.6. Let x=1x=1x=1x=1x=1 if gcd(n,p)≠1gcdâ¡(n,p)≠1gcd(n,p)!=1\operatorname{gcd}(\mathfrak{n}, p) \neq 1gcdâ¡(n,p)≠1 and let xxxxx be as in Lemma 6.4 if gcd(n,p)=1gcdâ¡(n,p)=1gcd(n,p)=1\operatorname{gcd}(\mathfrak{n}, p)=1gcdâ¡(n,p)=1. There exists an Rp/xΘ#Rp/xΘ#R_(p)//xTheta^(#)R_{p} / x \Theta^{\#}Rp/xΘ#-algebra WWWWW and a surjective RpRpR_(p)R_{p}Rp-algebra homomorphism φ:T→Wφ:T→Wvarphi:Trarr W\varphi: \mathbf{T} \rightarrow Wφ:T→W satisfying the following properties.
The structure map Rp/xΘ#→WRp/xΘ#→WR_(p)//xTheta^(#)rarr WR_{p} / x \Theta^{\#} \rightarrow WRp/xΘ#→W is an injection.
φ(TY)=ϵcyc k−1(l)+χ(l)φTY=ϵcyc k−1(l)+χ(l)varphi(T_(Y))=epsilon_("cyc ")^(k-1)(l)+chi(l)\varphi\left(T_{\mathfrak{Y}}\right)=\epsilon_{\text {cyc }}^{k-1}(\mathfrak{l})+\chi(\mathfrak{l})φ(TY)=ϵcyc k−1(l)+χ(l) for npnpnp\mathfrak{\mathfrak { n } p}np.
φ(Up)=1φUp=1varphi(U_(p))=1\varphi\left(U_{\mathfrak{p}}\right)=1φ(Up)=1 for p∣gcd(n,p)p∣gcdâ¡(n,p)p∣gcd(n,p)\mathfrak{p} \mid \operatorname{gcd}(\mathfrak{n}, p)p∣gcdâ¡(n,p).
If y∈Rpy∈Rpy inR_(p)y \in R_{p}y∈Rp and φ(U)y=0φ(U)y=0varphi(U)y=0\varphi(U) y=0φ(U)y=0 in WWWWW, then y∈(Θ#)y∈Θ#y in(Theta^(#))y \in\left(\Theta^{\#}\right)y∈(Θ#).
The idea of this theorem is the usual one: the homomorphism φφvarphi\varphiφ sends a Hecke
point: the operators UpUpU_(p)U_{\mathfrak{p}}Up for p∣p,p∤np∣p,p∤np∣p,p∤n\mathfrak{p} \mid p, \mathfrak{p} \nmid \mathfrak{n}p∣p,p∤n do not act as scalars, so a more involved argument is necessary. This explains why the ring WWWWW is not just Rp/xΘ#Rp/xΘ#R_(p)//xTheta^(#)R_{p} / x \Theta^{\#}Rp/xΘ#. The idea behind the last statement of the theorem is that the operator φ(U)φ(U)varphi(U)\varphi(U)φ(U) introduces a factor of xxxxx; If xyxyxyx yxy is divisible by xΘ#xΘ#xTheta^(#)x \Theta^{\#}xΘ# in RpRpR_(p)R_{p}Rp, then yyyyy is divisible by Θ#Θ#Theta^(#)\Theta^{\#}Θ# since xxxxx is a non-zero-divisor. This demonstrates the essential additional ingredient provided by the "higher congruence" modulo xΘ#xΘ#xTheta^(#)x \Theta^{\#}xΘ# rather than just modulo Θ#Θ#Theta^(#)\Theta^{\#}Θ#. See [17, THEOREM 8.23] for details.
6.5. Cusp forms to Galois representations
In this section we study the Galois representation attached to cusp forms that are congruent to Eisenstein series. Let mmm\mathfrak{m}m be the intersection of the finitely many maximal ideals of TTT\mathbf{T}T containing the kernel of φφvarphi\varphiφ. Put TmTmT_(m)\mathbf{T}_{\mathfrak{m}}Tm for the completion of TTT\mathbf{T}T with respect to mmm\mathfrak{m}m and K=Frac(Tm)K=Fracâ¡TmK=Frac(T_(m))K=\operatorname{Frac}\left(\mathbf{T}_{\mathfrak{m}}\right)K=Fracâ¡(Tm). Then KKKKK is a finite product of fields parameterized by the QpQpQ_(p)\mathbf{Q}_{p}Qp-Galois orbits of cuspidal newforms of weight kkkkk and level nnn\mathfrak{n}n, defined over the ring of integers in a finite extension of ZpZpZ_(p)\mathbf{Z}_{p}Zp, that are congruent to an Eisenstein series modulo the maximal ideal. As in [17, $9.2], the work of Hida and Wiles gives a Galois representation
(1) ρÏrho\rhoÏ is unramified outside npnpnp\mathfrak{n} pnp.
(2) For all primes l∤npl∤npl∤np\mathfrak{l} \nmid \mathfrak{n} pl∤np, the characteristic polynomial of ρÏrho\rhoÏ (Frob l)l)l)\mathfrak{l})l) is given by
where εcyc εcyc epsi_("cyc ")\varepsilon_{\text {cyc }}εcyc is the ppppp-adic cyclotomic character and ηQηQeta_(Q)\eta_{\mathrm{Q}}ηQ is an unramified character given by ηq(recq(ϖ−1))=Uqηqrecqâ¡Ï–−1=Uqeta_(q)(rec_(q)(Ï–^(-1)))=U_(q)\eta_{\mathrm{q}}\left(\operatorname{rec}_{\mathrm{q}}\left(\varpi^{-1}\right)\right)=U_{\mathrm{q}}ηq(recqâ¡(ϖ−1))=Uq, with ϖÏ–Ï–\varpiÏ– a uniformizer of Fq∗Fq∗F_(q)^(**)F_{\mathrm{q}}^{*}Fq∗.
Let I denote the kernel of φφvarphi\varphiφ extended to φ:Tm→Wφ:Tm→Wvarphi:T_(m)rarr W\varphi: \mathbf{T}_{\mathfrak{m}} \rightarrow Wφ:Tm→W. Reducing (6.13) modulo III\mathbf{I}I, and using ÄŒebotarev to extend from Frob bIbIb_(I)b_{\mathfrak{I}}bI to all σ∈GFσ∈GFsigma inG_(F)\sigma \in G_{F}σ∈GF, we see that the characteristic polynomial of ρ(σ)Ï(σ)rho(sigma)\rho(\sigma)Ï(σ) is congruent to (x−χ(σ))(x−ϵcyc(σ))(modI)(x−χ(σ))x−ϵcyc(σ)(modI)(x-chi(sigma))(x-epsilon_(cyc)(sigma))(modI)(x-\chi(\sigma))\left(x-\epsilon_{\mathrm{cyc}}(\sigma)\right)(\bmod \mathbf{I})(x−χ(σ))(x−ϵcyc(σ))(modI). In particular, if χ(σ)≢ϵcyc(σ)(modm)χ(σ)≢ϵcyc(σ)(modm)chi(sigma)≢epsilon_(cyc)(sigma)(modm)\chi(\sigma) \not \equiv \epsilon_{\mathrm{cyc}}(\sigma)(\bmod \mathfrak{m})χ(σ)≢ϵcyc(σ)(modm), then by Hensel's lemma ρ(σ)Ï(σ)rho(sigma)\rho(\sigma)Ï(σ) has distinct eigenvalues λ1,λ2∈Tmλ1,λ2∈Tmlambda_(1),lambda_(2)inT_(m)\lambda_{1}, \lambda_{2} \in \mathbf{T}_{\mathfrak{m}}λ1,λ2∈Tm such that λ1≡ϵcyc k−1(σ)(modI)λ1≡ϵcyc k−1(σ)(modI)lambda_(1)-=epsilon_("cyc ")^(k-1)(sigma)(mod I)\lambda_{1} \equiv \epsilon_{\text {cyc }}^{k-1}(\sigma)(\bmod I)λ1≡ϵcyc k−1(σ)(modI) and λ2≡χ(σ)(modI)λ2≡χ(σ)(modI)lambda_(2)-=chi(sigma)(modI)\lambda_{2} \equiv \chi(\sigma)(\bmod \mathbf{I})λ2≡χ(σ)(modI).
To define a convenient basis for ρÏrho\rhoÏ, we choose τ∈GFτ∈GFtau inG_(F)\tau \in G_{F}τ∈GF such that:
(1) τÏ„tau\tauÏ„ restricts to the complex conjugation of GGGGG,
(2) for each q∣pq∣pq∣pq \mid pq∣p, the eigenspace of ρ|GqÏGqrho|_(G_(q))\left.\rho\right|_{G_{q}}Ï|Gq projected to each factor of KKKKK is not stable under ρ(τ)Ï(Ï„)rho(tau)\rho(\tau)Ï(Ï„).
See [17, PRopositIoN 9.3] for the existence of such τÏ„tau\tauÏ„. Since p≠2p≠2p!=2p \neq 2p≠2, we have
It follows from the discussion above that the eigenvalues of ρ(τ)Ï(Ï„)rho(tau)\rho(\tau)Ï(Ï„) satisfy λ1≡ϵcyc(σ)(modI)λ1≡ϵcyc(σ)(modI)lambda_(1)-=epsilon_(cyc)(sigma)(modI)\lambda_{1} \equiv \epsilon_{\mathrm{cyc}}(\sigma)(\bmod \mathbf{I})λ1≡ϵcyc(σ)(modI) and λ2≡−1(modI)λ2≡−1(modI)lambda_(2)-=-1(modI)\lambda_{2} \equiv-1(\bmod \mathbf{I})λ2≡−1(modI). Fix the basis consisting of eigenvectors of ρ(τ)Ï(Ï„)rho(tau)\rho(\tau)Ï(Ï„), say
The second condition in the choice of τÏ„tau\tauÏ„ ensures that Aq,Cq∈K∗Aq,Cq∈K∗A_(q),C_(q)inK^(**)A_{\mathfrak{q}}, C_{\mathfrak{q}} \in K^{*}Aq,Cq∈K∗. Furthermore, equating the upper left-hand entries in (6.14) gives
(6.15)b(σ)=AqCq(a(σ)−χεcyck−1ηq−1(σ)) for all σ∈Gq(6.15)b(σ)=AqCqa(σ)−χεcyck−1ηq−1(σ) for all σ∈Gq{:(6.15)b(sigma)=(A_(q))/(C_(q))(a(sigma)-chiepsi_(cyc)^(k-1)eta_(q)^(-1)(sigma))quad" for all "sigma inG_(q):}\begin{equation*}
b(\sigma)=\frac{A_{\mathfrak{q}}}{C_{\boldsymbol{q}}}\left(a(\sigma)-\chi \varepsilon_{\mathrm{cyc}}^{k-1} \eta_{\mathrm{q}}^{-1}(\sigma)\right) \quad \text { for all } \sigma \in G_{\mathrm{q}} \tag{6.15}
\end{equation*}(6.15)b(σ)=AqCq(a(σ)−χεcyck−1ηq−1(σ)) for all σ∈Gq
6.6. Galois representations to Galois cohomology classes
We summarize [17, §9.3]. As explained above, we have
(6.16)a(σ)+d(σ)≡χ(σ)+ϵcyck−1(σ)(modI) for all σ∈GF(6.16)a(σ)+d(σ)≡χ(σ)+ϵcyck−1(σ)(modI) for all σ∈GF{:(6.16)a(sigma)+d(sigma)-=chi(sigma)+epsilon_(cyc)^(k-1)(sigma)(mod I)quad" for all "sigma inG_(F):}\begin{equation*}
a(\sigma)+d(\sigma) \equiv \chi(\sigma)+\epsilon_{\mathrm{cyc}}^{k-1}(\sigma)(\bmod I) \quad \text { for all } \sigma \in G_{F} \tag{6.16}
\end{equation*}(6.16)a(σ)+d(σ)≡χ(σ)+ϵcyck−1(σ)(modI) for all σ∈GF
Applying the same rule for τστσtau sigma\tau \sigmaτσ and noting that a(τσ)=λ1a(σ),d(στ)=λ2d(σ)a(τσ)=λ1a(σ),d(στ)=λ2d(σ)a(tau sigma)=lambda_(1)a(sigma),d(sigma tau)=lambda_(2)d(sigma)a(\tau \sigma)=\lambda_{1} a(\sigma), d(\sigma \tau)=\lambda_{2} d(\sigma)a(τσ)=λ1a(σ),d(στ)=λ2d(σ), we find
Solving the congruences (6.16) and (6.17) and once again using the fact that ϵcyck−1(τ)≢−1ϵcyck−1(Ï„)≢−1epsilon_(cyc)^(k-1)(tau)≢-1\epsilon_{\mathrm{cyc}}^{k-1}(\tau) \not \equiv-1ϵcyck−1(Ï„)≢−1(modm)(modm)(modm)(\bmod \mathfrak{m})(modm) since p≠2p≠2p!=2p \neq 2p≠2, we find that a(σ),d(σ)∈Tma(σ),d(σ)∈Tma(sigma),d(sigma)inT_(m)a(\sigma), d(\sigma) \in \mathbf{T}_{\mathfrak{m}}a(σ),d(σ)∈Tm and
(6.18)a(σ)≡εcyck−1(σ)(modI),d(σ)≡χ(σ)(modI) for all σ∈GF(6.18)a(σ)≡εcyck−1(σ)(modI),d(σ)≡χ(σ)(modI) for all σ∈GF{:(6.18)a(sigma)-=epsi_(cyc)^(k-1)(sigma)(modI)","quad d(sigma)-=chi(sigma)(modI)quad" for all "sigma inG_(F):}\begin{equation*}
a(\sigma) \equiv \varepsilon_{\mathrm{cyc}}^{k-1}(\sigma)(\bmod \mathbf{I}), \quad d(\sigma) \equiv \chi(\sigma)(\bmod \mathbf{I}) \quad \text { for all } \sigma \in G_{F} \tag{6.18}
\end{equation*}(6.18)a(σ)≡εcyck−1(σ)(modI),d(σ)≡χ(σ)(modI) for all σ∈GF
Let BBBBB be the TmTmT_(m)\mathbf{T}_{\mathfrak{m}}Tm submodule of KKKKK generated by {b(σ):σ∈GF}∪{AqCq:q∈b(σ):σ∈GF∪AqCq:q∈{b(sigma):sigma inG_(F)}uu{(A_(q))/(C_(q)):q in:}\left\{b(\sigma): \sigma \in G_{F}\right\} \cup\left\{\frac{A_{\mathfrak{q}}}{C_{\boldsymbol{q}}}: q \in\right.{b(σ):σ∈GF}∪{AqCq:q∈Σ∖S∞}Σ∖S∞{: Sigma\\S_(oo)}\left.\Sigma \backslash S_{\infty}\right\}Σ∖S∞}. We have ρ(σσ′)=ρ(σ)ρ(σ′)Ïσσ′=Ï(σ)Ïσ′rho(sigmasigma^('))=rho(sigma)rho(sigma^('))\rho\left(\sigma \sigma^{\prime}\right)=\rho(\sigma) \rho\left(\sigma^{\prime}\right)Ï(σσ′)=Ï(σ)Ï(σ′) for σ,σ′∈GFσ,σ′∈GFsigma,sigma^(')inG_(F)\sigma, \sigma^{\prime} \in G_{F}σ,σ′∈GF. Equating the upper right entries and using equation (6.18), we obtain
Let mmmmm be am integer such that k≡1(mod(p−1)pm)k≡1mod(p−1)pmk-=1(mod(p-1)p^(m))k \equiv 1\left(\bmod (p-1) p^{m}\right)k≡1(mod(p−1)pm). Let IqIqI_(q)I_{\mathfrak{q}}Iq denote the inertia subgroup of GFGFG_(F)G_{F}GF of a prime q. Put B1B1B_(1)B_{1}B1 for the TmTmT_(m)\mathbf{T}_{\mathfrak{m}}Tm-submodule of BBBBB generated by
IB∪pmB∪{b(σ):σ∈Iq for q∣p,q∉Σ}IB∪pmB∪b(σ):σ∈Iq for q∣p,q∉ΣIB uup^(m)B uu{b(sigma):sigma inI_(q)" for "q∣p,q!in Sigma}\mathbf{I} B \cup p^{m} B \cup\left\{b(\sigma): \sigma \in I_{\mathrm{q}} \text { for } \mathfrak{q} \mid p, q \notin \Sigma\right\}IB∪pmB∪{b(σ):σ∈Iq for q∣p,q∉Σ}
Define B¯=B/B1B¯=B/B1bar(B)=B//B_(1)\bar{B}=B / B_{1}B¯=B/B1. Equation (6.19) then gives that κ(σ)=χ−1(σ)b(σ)κ(σ)=χ−1(σ)b(σ)kappa(sigma)=chi^(-1)(sigma)b(sigma)\kappa(\sigma)=\chi^{-1}(\sigma) b(\sigma)κ(σ)=χ−1(σ)b(σ) is a cocyle defining a cohomology class [κ][κ][kappa][\kappa][κ] in H1(GF,B¯(χ−1))H1GF,B¯χ−1H^(1)(G_(F),( bar(B))(chi^(-1)))H^{1}\left(G_{F}, \bar{B}\left(\chi^{-1}\right)\right)H1(GF,B¯(χ−1)) satisfying the following local properties:
(1) As ρÏrho\rhoÏ is unramified at Υ∤πpΥ∤πpΥ∤pi p\mathfrak{\Upsilon} \nmid \pi pΥ∤πp, so is the class [κ][κ][kappa][\kappa][κ].
(2) As B¯B¯bar(B)\bar{B}B¯ is pro- ppppp, the class [κ][κ][kappa][\kappa][κ] is at most tamely ramified at any prime l∣nl∣nl∣n\mathfrak{l} \mid \mathfrak{n}l∣n not above ppppp.
(3) It is proven in [17,84.1][17,84.1][17,84.1][17,84.1][17,84.1] that we may assume Σ′Σ′Sigma^(')\Sigma^{\prime}Σ′ does not contain any primes above ppppp. Thus [κ][κ][kappa][\kappa][κ] is at most tamely ramified at all primes in Σ′Σ′Sigma^(')\Sigma^{\prime}Σ′.
(4) By the definition of B1B1B_(1)B_{1}B1, where we have included b(Iq)bIqb(I_(q))b\left(I_{q}\right)b(Iq) for primes q∣p,q∉Σq∣p,q∉Σq∣p,q!in Sigmaq \mid p, q \notin \Sigmaq∣p,q∉Σ, the class [κ][κ][kappa][\kappa][κ] is unramified at such qqqqq.
(5) Equation (6.15) implies that [κ][κ][kappa][\kappa][κ] is locally trivial at finite primes in ΣΣSigma\SigmaΣ. As ppppp is odd, [κ][κ][kappa][\kappa][κ] is locally trivial at archimedian places [17, PROPOSITION 9.5].
6.7. Galois cohomology classes to class groups
The Galois cohomology class [κ][κ][kappa][\kappa][κ] satisfies the conditions listed after equation (6.2) and hence gives a surjection
It is therefore enough to prove that FittRp(B¯)⊂(Θ#)FittRpâ¡(B¯)⊂Θ#Fitt_(R_(p))( bar(B))sub(Theta^(#))\operatorname{Fitt}_{R_{p}}(\bar{B}) \subset\left(\Theta^{\#}\right)FittRpâ¡(B¯)⊂(Θ#). Typically in Ribet's method, one argues that the fractional ideal BBBBB is a faithful TmTmT_(m)\mathbf{T}_{\mathfrak{m}}Tm-module, and hence the Fitting ideal of B/IBB/IBB//IBB / \mathbf{I} BB/IB is contained in III\mathbf{I}I. However, our module B¯B¯bar(B)\bar{B}B¯ is more complicated than B/IBB/IBB//IBB / \mathbf{I} BB/IB, so we proceed as follows. Using equation (6.15), we show that any element in FittRp(B¯)FittRpâ¡(B¯)Fitt_(R_(p))( bar(B))\operatorname{Fitt}_{R_{p}}(\bar{B})FittRpâ¡(B¯) is annihilated by φ(U)φ(U)varphi(U)\varphi(U)φ(U) for the operator UUUUU from Theorem 6.6. The final assertion of this theorem then implies that FittRp(B¯)FittRpâ¡(B¯)Fitt_(R_(p))( bar(B))\operatorname{Fitt}_{R_{p}}(\bar{B})FittRpâ¡(B¯) contained in (Θ#)Θ#(Theta^(#))\left(\Theta^{\#}\right)(Θ#). See [17, §9.5] for details.
This concludes our summary of the proof of Theorem 5.6.
7. EXPLICIT FORMULA FOR BRUMER-STARK UNITS
In this final section of the paper, we discuss the first author's explicit formula for Brumer-Stark units as mentioned in §4.3§4.3§4.3\S 4.3§§4.3. The conjecture in the case that FFFFF is a real quadratic field was studied in [13], and the general case was studied in [15]. Here we consider an arbitrary totally real field FFFFF, but to simplify formulas we assume that the rational prime ppppp is inert in FFFFF. Furthermore, we let HHHHH be the narrow ray class field of some conductor n⊂OFn⊂OFnsubO_(F)\mathfrak{n} \subset O_{F}n⊂OF and assume that p≡1(modn)p≡1(modn)p-=1(modn)p \equiv 1(\bmod \mathfrak{n})p≡1(modn). This ensures that the prime p=pOFp=pOFp=pO_(F)\mathfrak{p}=p O_{F}p=pOF splits completely in HHHHH. Fix a prime PPP\mathfrak{P}P of HHHHH above ppppp. We fix S⊃S∞∪Sram ={v∣n∞}S⊃S∞∪Sram ={v∣n∞}S supS_(oo)uuS_("ram ")={v∣noo}S \supset S_{\infty} \cup S_{\text {ram }}=\{v \mid \mathfrak{n} \infty\}S⊃S∞∪Sram ={v∣n∞}. We also fix a prime ideal l⊂OFl⊂OFlsubO_(F)\mathfrak{l} \subset O_{F}l⊂OF such that NY=ℓ>n+1NY=â„“>n+1NY=â„“ > n+1\mathrm{N} \mathfrak{Y}=\ell>n+1NY=â„“>n+1 is a prime integer and let T={l}T={l}T={l}T=\{\mathfrak{l}\}T={l}.
In this setting, we will present a ppppp-adic analytic formula for the image of the BrumerStark unit up∈H∗up∈H∗u_(p)inH^(**)u_{\mathfrak{p}} \in H^{*}up∈H∗ in HP∗≅Fp∗HP∗≅Fp∗H_(P)^(**)~=F_(p)^(**)H_{\mathfrak{P}}^{*} \cong F_{\mathfrak{p}}^{*}HP∗≅Fp∗. The most general, conceptually satisfying, and theoretically useful form of this conjecture uses the Eisenstein cocycle. This is a class in the (n−1)(n−1)(n-1)(n-1)(n−1) st cohomology of GLn(Z)GLn(Z)GL_(n)(Z)\mathrm{GL}_{n}(\mathbf{Z})GLn(Z) that has many avatars studied by several authors (see [2,3,9,10,13[2,3,9,10,13[2,3,9,10,13[2,3,9,10,13[2,3,9,10,13,
20, 45]). In this paper, we avoid defining the Eisenstein cocycle and present instead the more explicit and down to earth version of the conjectural formula for upupu_(p)u_{\mathfrak{p}}up stated in [15].
7.1. Shintani's method
Fixing an ordering of the nnnnn real embeddings of FFFFF yields an map F↪RnF↪RnF↪R^(n)F \hookrightarrow \mathbf{R}^{n}F↪Rn such that the image of any fractional ideal is a cocompact lattice. We let F∗F∗F^(**)F^{*}F∗ act on RnRnR^(n)\mathbf{R}^{n}Rn by composing this embedding with componentwise multiplication and denote the action by ∗∗***∗.
Let v1,…,vr∈(R>0)n,1≤r≤nv1,…,vr∈R>0n,1≤r≤nv_(1),dots,v_(r)in(R^( > 0))^(n),1 <= r <= nv_{1}, \ldots, v_{r} \in\left(\mathbf{R}^{>0}\right)^{n}, 1 \leq r \leq nv1,…,vr∈(R>0)n,1≤r≤n, be vectors in the totally positive orthant that are linearly independent over RRR\mathbf{R}R. The corresponding simplicial cone is defined by
Suppose now r=nr=nr=nr=nr=n. We will define a certain union of C(v1,…,vn)Cv1,…,vnC(v_(1),dots,v_(n))C\left(v_{1}, \ldots, v_{n}\right)C(v1,…,vn) and some of its boundary faces that we call the Colmez closure. Write
For each nonempty subset J⊂{1,…,n}J⊂{1,…,n}J sub{1,dots,n}J \subset\{1, \ldots, n\}J⊂{1,…,n}, we say that JJJJJ is positive if qi>0qi>0q_(i) > 0q_{i}>0qi>0 for all i∉Ji∉Ji!in Ji \notin Ji∉J. The Colmez closure of C(v1,…,vn)Cv1,…,vnC(v_(1),dots,v_(n))C\left(v_{1}, \ldots, v_{n}\right)C(v1,…,vn) is defined by:
Definition 7.1. A signed fundamental domain for the action of E(n)E(n)E(n)E(\mathfrak{n})E(n) on (R>0)nR>0n(R^( > 0))^(n)\left(\mathbf{R}^{>0}\right)^{n}(R>0)n is by definition a formal linear combination D=∑iaiCiD=∑i aiCiD=sum_(i)a_(i)C_(i)D=\sum_{i} a_{i} C_{i}D=∑iaiCi of simplicial cones CiCiC_(i)C_{i}Ci with ai∈Zai∈Za_(i)inZa_{i} \in \mathbf{Z}ai∈Z such that
∑u∈E(n)∑iai1Ci(u∗x)=1∑u∈E(n) ∑i ai1Ci(u∗x)=1sum_(u in E(n))sum_(i)a_(i)1_(C_(i))(u**x)=1\sum_{u \in E(\mathfrak{n})} \sum_{i} a_{i} \mathbf{1}_{C_{i}}(u * x)=1∑u∈E(n)∑iai1Ci(u∗x)=1
for all x∈(R>0)nx∈R>0nx in(R_( > 0))^(n)x \in\left(\mathbf{R}_{>0}\right)^{n}x∈(R>0)n.
Fix an ordered basis {ϵ1,…,ϵn−1}ϵ1,…,ϵn−1{epsilon_(1),dots,epsilon_(n-1)}\left\{\epsilon_{1}, \ldots, \epsilon_{n-1}\right\}{ϵ1,…,ϵn−1} for E(n)E(n)E(n)E(\mathfrak{n})E(n). Define the orientation
where ϵijϵijepsilon_(ij)\epsilon_{i j}ϵij denotes the jjjjj th coordinate of ϵiϵiepsilon_(i)\epsilon_{i}ϵi. For each permutation σ∈Sn−1σ∈Sn−1sigma inS_(n-1)\sigma \in S_{n-1}σ∈Sn−1 let
The following result was proven independently by Diaz y Diaz-Friedman [22] and Charollois-Dasgupta-Greenberg [10, THEOREM 1.5], generalizing the result of Colmez [11] in the case that all wσ=1wσ=1w_(sigma)=1w_{\sigma}=1wσ=1.
is a signed fundamental domain for the action of E(n)E(n)E(n)E(\mathfrak{n})E(n) on (R>0)nR>0n(R^( > 0))^(n)\left(\mathbf{R}^{>0}\right)^{n}(R>0)n.
7.2. The formula
Throughout this section assume that ppppp is odd. Recall that T={l}T={l}T={l}T=\{\mathfrak{l}\}T={l}. Let bbb\mathfrak{b}b be a fractional ideal that is relatively prime to nlnlnl\mathfrak{n l}nl. Let D=∑aiCiD=∑aiCiD=suma_(i)C_(i)D=\sum a_{i} C_{i}D=∑aiCi be the signed fundamental domain for the action of E(n)E(n)E(n)E(\mathfrak{n})E(n) on (R>0)nR>0n(R^( > 0))^(n)\left(\mathbf{R}^{>0}\right)^{n}(R>0)n given in Theorem 7.2. We use all this data to define a ZZZ\mathbf{Z}Z-valued measure μμmu\muμ on OpOpO_(p)O_{p}Op, the ppppp-adic completion of OFOFO_(F)O_{F}OF. Fix an element z∈b−1z∈b−1z inb^(-1)z \in \mathfrak{b}^{-1}z∈b−1 such that z≡1(modn)z≡1(modn)z-=1(modn)z \equiv 1(\bmod \mathfrak{n})z≡1(modn). For each compact open set U⊂OpU⊂OpU subO_(p)U \subset O_{p}U⊂Op, define the Shintani zeta function
Shintani proved that this sum converges for ℜ(s)ℜ(s)ℜ(s)\Re(s)ℜ(s) large enough and extends to a meromorphic function on CCC\mathbf{C}C. Define
μb(U)=ζ(b,U,D,0)−ℓ⋅ζ(bL−1,U,D,0)μb(U)=ζ(b,U,D,0)−ℓ⋅ζbL−1,U,D,0mu_(b)(U)=zeta(b,U,D,0)-â„“*zeta(bL^(-1),U,D,0)\mu_{\mathfrak{b}}(U)=\zeta(\mathfrak{b}, U, D, 0)-\ell \cdot \zeta\left(\mathfrak{b} \mathfrak{L}^{-1}, U, D, 0\right)μb(U)=ζ(b,U,D,0)−ℓ⋅ζ(bL−1,U,D,0)
Using Shintani's formulas, one may show:
Theorem 7.3 ([15, PRoPosition 3.12]). For every compact open U⊂OpU⊂OpU subO_(p)U \subset O_{p}U⊂Op, we have μb(U)∈Zμb(U)∈Zmu_(b)(U)inZ\mu_{\mathfrak{b}}(U) \in \mathbf{Z}μb(U)∈Z.
We may now state our conjectural exact formula for the Brumer-Stark unit upupu_(p)u_{p}up and all of its conjugates over FFFFF. Write
(7.2)up(b)an=pζs,T(σb)ψOp∗xdμb(x)∈Fp∗(7.2)up(b)an=pζs,TσbψOp∗xdμb(x)∈Fp∗{:(7.2)u_(p)(b)^(an)=p^(zeta s,T(sigma_(b)))psi_(O_(p)^(**))xdmu_(b)(x)inF_(p)^(**):}\begin{equation*}
u_{p}(\mathfrak{b})^{\mathrm{an}}=p^{\zeta s, T\left(\sigma_{\mathfrak{b}}\right)} \psi_{O_{p}^{*}} x d \mu_{\mathfrak{b}}(x) \in F_{p}^{*} \tag{7.2}
\end{equation*}(7.2)up(b)an=pζs,T(σb)ψOp∗xdμb(x)∈Fp∗
Here the crossed integral is a multiplicative integral in the sense of Darmon [12] and can be expressed as a limit of Riemann products:
fOp∗xdμb(x):=limm→∞∏a∈(Op/pm)∗aμF(a+pmOp)fOp∗xdμb(x):=limm→∞ âˆa∈Op/pm∗ aμFa+pmOpf_(O_(p)^(**))xdmu_(b)(x):=lim_(m rarr oo)prod_(a in(O_(p)//p^(m))^(**))a^(mu_(F)(a+p^(m)O_(p)))\mathcal{f}_{O_{p}^{*}} x d \mu_{\mathfrak{b}}(x):=\lim _{m \rightarrow \infty} \prod_{a \in\left(O_{p} / p^{m}\right)^{*}} a^{\mu_{\mathfrak{F}}\left(a+p^{m} O_{p}\right)}fOp∗xdμb(x):=limm→∞âˆa∈(Op/pm)∗aμF(a+pmOp)
Write σb∈Gσb∈Gsigma_(b)in G\sigma_{\mathfrak{b}} \in Gσb∈G for the Frobenius associated to bbb\mathfrak{b}b. In [15, THEOREM 5.15] we prove that up(b)anup(b)anu_(p)(b)^(an)u_{p}(\mathfrak{b})^{\mathrm{an}}up(b)an depends only on the image of bbb\mathfrak{b}b in the narrow ray class group of conductor nnn\mathfrak{n}n, i.e., on σb∈Gσb∈Gsigma_(b)in G\sigma_{\mathfrak{b}} \in Gσb∈G (at least up to a root of unity in Fp∗Fp∗F_(p)^(**)F_{p}^{*}Fp∗ ).
Conjecture 7.4. We have σb(up)=up(b)anσbup=up(b)ansigma_(b)(u_(p))=u_(p)(b)^(an)\sigma_{\mathfrak{b}}\left(u_{\mathfrak{p}}\right)=u_{p}(\mathfrak{b})^{\mathrm{an}}σb(up)=up(b)an in Fp∗Fp∗F_(p)^(**)F_{p}^{*}Fp∗.
The expression (7.2) can be computed to high ppppp-adic precision on a computer. See [23] for tables of narrow Hilbert class fields of real quadratic fields determined using this formula.
It is convenient to have an invariant that also satisfies up(bq)an=(up(b)an)−1up(bq)an=up(b)an−1u_(p)(bq)^(an)=(u_(p)(b)^(an))^(-1)u_{p}(\mathfrak{b q})^{\mathrm{an}}=\left(u_{p}(\mathfrak{b})^{\mathrm{an}}\right)^{-1}up(bq)an=(up(b)an)−1 if qqq\mathfrak{q}q is a prime such that σq=cσq=csigma_(q)=c\sigma_{q}=cσq=c. Conjecture 7.4 would imply such a formula, but it is unclear whether this purely analytic statement can be proved unconditionally. To this end, we fix qqqqq such that σq˙=cσqË™=csigma_(q^(Ë™))=c\sigma_{\dot{q}}=cσqË™=c and define
unconditionally, and we expect to have vp(b)an=up(b)anvp(b)an=up(b)anv_(p)(b)^(an)=u_(p)(b)^(an)v_{p}(\mathfrak{b})^{\mathrm{an}}=u_{p}(\mathfrak{b})^{\mathrm{an}}vp(b)an=up(b)an. The following is therefore a slightly easier form of Conjecture 7.4 to study.
Conjecture 7.5. We have σb(up)=vp(b)anσbup=vp(b)ansigma_(b)(u_(p))=v_(p)(b)^(an)\sigma_{\mathfrak{b}}\left(u_{\mathfrak{p}}\right)=v_{p}(\mathfrak{b})^{\mathrm{an}}σb(up)=vp(b)an in F^p∗F^p∗hat(F)_(p)^(**)\hat{F}_{p}^{*}F^p∗.
7.3. Horizontal Iwasawa theory
We now discuss the relationship between Gross's tower of fields conjecture (Conjecture 4.4) and our conjectural exact formula for Brumer-Stark units. Our goal is to prove:
Theorem 7.6. Assume that ppppp is odd. Gross's conjecture implies Conjecture 7.5.
In this exposition we have assumed that the odd prime ppppp is inert in FFFFF and that p=pOFp=pOFp=pO_(F)p=p O_{F}p=pOF. In the case of general ppp\mathfrak{p}p, one must still assume that ppppp is odd and unramified in FFFFF in the statement of Theorem 7.6.
The abelian extensions L/FL/FL//FL / FL/F to which we can apply Gross's conjecture (with S′=S∪{p}S′=S∪{p}S^(')=S uu{p}S^{\prime}=S \cup\{\mathfrak{p}\}S′=S∪{p} ) as in Conjecture 4.4) are those that contain HHHHH and are unramified outside S′∞S′∞S^(')ooS^{\prime} \inftyS′∞. Let FS′FS′F_(S^('))F_{S^{\prime}}FS′ denote the maximal abelian extension of FFFFF unramified outside S′∞S′∞S^(')ooS^{\prime} \inftyS′∞. The reciprocity map of class field theory yields an explicit description of Gal(FS′/H)Galâ¡FS′/HGal(F_(S^('))//H)\operatorname{Gal}\left(F_{S^{\prime}} / H\right)Galâ¡(FS′/H). For each finite v∈S′v∈S′v inS^(')v \in S^{\prime}v∈S′, let Uv,n⊂Ov∗Uv,n⊂Ov∗U_(v,n)subO_(v)^(**)U_{v, \mathfrak{n}} \subset O_{v}^{*}Uv,n⊂Ov∗ denote the subgroup of elements congruent to 1 modulo nOvnOvnO_(v)\mathfrak{n} O_{v}nOv (so Uf,n=Ov∗Uf,n=Ov∗U_(f,n)=O_(v)^(**)U_{f, \mathfrak{n}}=O_{v}^{*}Uf,n=Ov∗ for v∤nv∤nv∤nv \nmid \mathfrak{n}v∤n ). Define O∗=∏v∈S′∖S∞Uv,nO∗=âˆv∈S′∖S∞ Uv,nO^(**)=prod_(v inS^(')\\S_(oo))U_(v,n)\mathbf{O}^{*}=\prod_{v \in S^{\prime} \backslash S_{\infty}} U_{v, \mathfrak{n}}O∗=âˆv∈S′∖S∞Uv,n. Then
where E(n)¯E(n)¯bar(E(n))\overline{E(\mathfrak{n})}E(n)¯ denotes the topological closure of E(n)E(n)E(n)E(\mathfrak{n})E(n) embedded diagonally in O∗O∗O^(**)\mathbf{O}^{*}O∗.
For each finite extension L⊂FS′L⊂FS′L subF_(S^('))L \subset F_{S^{\prime}}L⊂FS′ containing HHHHH, if we write Γ=Gal(L/H)Γ=Galâ¡(L/H)Gamma=Gal(L//H)\Gamma=\operatorname{Gal}(L / H)Γ=Galâ¡(L/H), then (4.7) yields a formula for recG(up)recGâ¡uprec_(G)(u_(p))\operatorname{rec}_{G}\left(u_{p}\right)recGâ¡(up) in I/I2≅Z[G]⊗ΓI/I2≅Z[G]⊗ΓI//I^(2)~=Z[G]ox GammaI / I^{2} \cong \mathbf{Z}[G] \otimes \GammaI/I2≅Z[G]⊗Γ. Under this isomorphism, the coefficient of σb−1σb−1sigma_(b)^(-1)\sigma_{\mathfrak{b}}^{-1}σb−1 is just the image of recp(σb(up))recpâ¡Ïƒbuprec_(p)(sigma_(b)(u_(p)))\operatorname{rec}_{\mathfrak{p}}\left(\sigma_{\mathfrak{b}}\left(u_{p}\right)\right)recpâ¡(σb(up)) in ΓΓGamma\GammaΓ. Taking the inverse limit over all H⊂L⊂FS′H⊂L⊂FS′H sub L subF_(S^('))H \subset L \subset F_{S^{\prime}}H⊂L⊂FS′ therefore gives an equality for
(σb(up),1,1,…,1) in O/E(n)¯Ïƒbup,1,1,…,1 in O/E(n)¯(sigma_(b)(u_(p)),1,1,dots,1)quad" in "O// bar(E(n))\left(\sigma_{\mathfrak{b}}\left(u_{p}\right), 1,1, \ldots, 1\right) \quad \text { in } \mathbf{O} / \overline{E(\mathfrak{n})}(σb(up),1,1,…,1) in O/E(n)¯
Here we have written Op∗Op∗O_(p)^(**)O_{p}^{*}Op∗ as the first component of OOO\mathbf{O}O.
The next key point is that the constructions of Section 7.2 can be repeated to provide a measure μb,Oμb,Omu_(b,O)\mu_{\mathfrak{b}, \mathbf{O}}μb,O on O=∏v∈S′∖S∞OvO=âˆv∈S′∖S∞ OvO=prod_(v inS^(')\\S_(oo))O_(v)\mathbf{O}=\prod_{v \in S^{\prime} \backslash S_{\infty}} O_{v}O=âˆv∈S′∖S∞Ov extending the measure μbμbmu_(b)\mu_{\mathfrak{b}}μb on OpOpO_(p)O_{p}Op. It is not hard to check that the restriction of μb,oμb,omu_(b,o)\mu_{\mathfrak{b}, \mathbf{o}}μb,o to O∗O∗O^(**)\mathbf{O}^{*}O∗, pushed forward to O∗/E(n)¯O∗/E(n)¯O^(**)// bar(E(n))\mathbf{O}^{*} / \overline{E(\mathfrak{n})}O∗/E(n)¯, is precisely the measure that recovers the values of the partial zeta functions of the abelian extensions LLLLL contained in FS′FS′F_(S^('))F_{S^{\prime}}FS′. These are exactly the values appearing in Gross's conjecture. In other words, Gross's conjecture for the set S′S′S^(')S^{\prime}S′ is equivalent to
(7.4)(σb(up),1,1,…,1)⋅p−ζS,T(σb)=∫O∗xdμb,O(x) in O/E(n)¯(7.4)σbup,1,1,…,1â‹…p−ζS,Tσb=∫O∗ xdμb,O(x) in O/E(n)¯{:(7.4)(sigma_(b)(u_(p)),1,1,dots,1)*p^(-zeta_(S,T)(sigma_(b)))=int_(O^(**))xdmu_(b,O)(x)quad" in "O// bar(E(n)):}\begin{equation*}
\left(\sigma_{\mathfrak{b}}\left(u_{\mathfrak{p}}\right), 1,1, \ldots, 1\right) \cdot p^{-\zeta_{S, T}\left(\sigma_{\mathfrak{b}}\right)}=\int_{\mathbf{O}^{*}} x d \mu_{\mathfrak{b}, \mathbf{O}}(x) \quad \text { in } \mathbf{O} / \overline{E(\mathfrak{n})} \tag{7.4}
\end{equation*}(7.4)(σb(up),1,1,…,1)⋅p−ζS,T(σb)=∫O∗xdμb,O(x) in O/E(n)¯
See [15, PROPOSITION 3.4]. The next important calculation ([15, THEOREM 3.22]) is that
(7.5)pζS,T(σb)∫O∗xdμb,o(x)=(up(b)an,1,1,…,1)(7.5)pζS,Tσb∫O∗ xdμb,o(x)=up(b)an,1,1,…,1{:(7.5)p^(zeta_(S,T)(sigma_(b)))int_(O^(**))xdmu_(b,o)(x)=(u_(p)(b)^(an),1,1,dots,1):}\begin{equation*}
p^{\zeta_{S, T}\left(\sigma_{\mathfrak{b}}\right)} \int_{\mathbf{O}^{*}} x d \mu_{\mathfrak{b}, \mathbf{o}}(x)=\left(u_{p}(\mathfrak{b})^{\mathrm{an}}, 1,1, \ldots, 1\right) \tag{7.5}
\end{equation*}(7.5)pζS,T(σb)∫O∗xdμb,o(x)=(up(b)an,1,1,…,1)
The first component of this is simply the compatibility of the constructions of μbμbmu_(b)\mu_{\mathfrak{b}}μb and μb,oμb,omu_(b,o)\mu_{\mathfrak{b}, \mathbf{o}}μb,o; the interesting part of the computation is the 1's in the components away from ppppp. Equations (7.4) and (7.5) combine to yield that the ratio σb(up)/up(b)an σbup/up(b)an sigma_(b)(u_(p))//u_(p)(b)^("an ")\sigma_{\mathfrak{b}}\left(u_{p}\right) / u_{p}(\mathfrak{b})^{\text {an }}σb(up)/up(b)an lies in the group
since c(up)=up−1cup=up−1c(u_(p))=u_(p)^(-1)c\left(u_{p}\right)=u_{p}^{-1}c(up)=up−1.
The final trick, inspired by the method of Taylor-Wiles, is to consider certain enlarged sets SQ=S∪QSQ=S∪QS_(Q)=S uu QS_{Q}=S \cup QSQ=S∪Q for a well-chosen finite set of auxiliary primes QQQQQ. Let us compare the Brumer-Stark units for SSSSS and SQSQS_(Q)S_{Q}SQ, denoted upupu_(p)u_{p}up and up(SQ)upSQu_(p)(S_(Q))u_{p}\left(S_{Q}\right)up(SQ), respectively. The defining property (4.1) shows that
up(SQ)=upz, where z=∏q∈Q(1−σq−1)∈Z[G]upSQ=upz, where z=âˆq∈Q 1−σq−1∈Z[G]u_(p)(S_(Q))=u_(p)^(z),quad" where "z=prod_(q in Q)(1-sigma_(q)^(-1))inZ[G]u_{p}\left(S_{Q}\right)=u_{p}^{z}, \quad \text { where } z=\prod_{q \in Q}\left(1-\sigma_{q}^{-1}\right) \in \mathbf{Z}[G]up(SQ)=upz, where z=âˆq∈Q(1−σq−1)∈Z[G]
In particular, if we choose the q∈Qq∈Qq in Qq \in Qq∈Q such that σq=cσq=csigma_(q)=c\sigma_{\mathfrak{q}}=cσq=c, the complex conjugation of GGGGG, then up(SQ)=up22∉QupSQ=up22∉Qu_(p)(S_(Q))=u_(p)^(2^(2!in Q))u_{p}\left(S_{Q}\right)=u_{p}^{2^{2 \notin Q}}up(SQ)=up22∉Q. Using (7.3), one can similarly show that vp(SQ,b)=vp(b222QvpSQ,b=vpb222Qv_(p)(S_(Q),b)=v_(p)(b2^(2^(2Q)):}v_{p}\left(S_{Q}, \mathfrak{b}\right)=v_{p}\left(\mathfrak{b} 2^{2^{2 Q}}\right.vp(SQ,b)=vp(b222Q. Now (7.6) for SQSQS_(Q)S_{Q}SQ implies that
(σb(up)/vp(b)an)2#Q∈D(SQ), so σb(up)/vp(b)an∈D(SQ)σbup/vp(b)an2#Q∈DSQ, so σbup/vp(b)an∈DSQ(sigma_(b)(u_(p))//v_(p)(b)^(an))^(2^(#Q))in D(S_(Q)),quad" so "sigma_(b)(u_(p))//v_(p)(b)^(an)in D(S_(Q))\left(\sigma_{\mathfrak{b}}\left(u_{p}\right) / v_{p}(\mathfrak{b})^{\mathrm{an}}\right)^{2^{\# Q}} \in D\left(S_{Q}\right), \quad \text { so } \sigma_{\mathfrak{b}}\left(u_{p}\right) / v_{p}(\mathfrak{b})^{\mathrm{an}} \in D\left(S_{Q}\right)(σb(up)/vp(b)an)2#Q∈D(SQ), so σb(up)/vp(b)an∈D(SQ)
since p≠2p≠2p!=2p \neq 2p≠2.
To conclude the proof of Theorem 7.6, one shows using the ÄŒebotarev Density Theorem that one can choose the sets QQQQQ to force D(SQ)DSQD(S_(Q))D\left(S_{Q}\right)D(SQ) as small as desired (i.e., the intersection of D(SQ)DSQD(S_(Q))D\left(S_{Q}\right)D(SQ) over all possible QQQQQ is trivial). See [15, LEMMA 5.17] for details.
7.4. The Greenberg-Stevens LLL\mathscr{L}L-invariant
We briefly summarize our proof of the ppppp-part of Gross's conjecture (Theorem 4.5), which as just explained implies our explicit formula for Brumer-Stark units given in Conjecture 7.5.
The work of Greenberg and Stevens [24] was a seminal breakthrough in the study of trivial zeroes of ppppp-adic LLLLL-functions. Their perspective was highly influential in [16], where the rank one ppppp-adic Gross-Stark conjecture was interpreted as the equality of an algebraic LLLLL-invariant Lalg Lalg L_("alg ")\mathscr{L}_{\text {alg }}Lalg and an analytic LLLLL-invariant Lan Lan L_("an ")\mathscr{L}_{\text {an }}Lan . The analytic LLL\mathscr{L}L-invariant is the ratio of the leading term of the ppppp-adic LLLLL-function at s=0s=0s=0s=0s=0 to its classical counterpart,
The algebraic LLLLL-invariant is the ratio of the ppppp-adic logarithm and valuation of the χ−1χ−1chi^(-1)\chi^{-1}χ−1 component of the Brumer-Stark unit,
There is no difficulty in defining the ratios (7.7) and (7.8), since the quantities live in a ppppp-adic field and the denominators are nonzero. The analogue of this situation for Gross's Conjecture 4.4 is more delicate. The role of the ppppp-adic LLLLL-function is played by the Stickelberger element ΘL:=ΘS′,T(L/F,0)∈Z[g]ΘL:=ΘS′,T(L/F,0)∈Z[g]Theta_(L):=Theta_(S^('),T)(L//F,0)inZ[g]\Theta_{L}:=\Theta_{S^{\prime}, T}(L / F, 0) \in \mathbf{Z}[g]ΘL:=ΘS′,T(L/F,0)∈Z[g], and the analogue of the derivative at 0 is played by the image of ΘLΘLTheta_(L)\Theta_{L}ΘL in I/I2I/I2I//I^(2)I / I^{2}I/I2. The role of the classical LLLLL-function is played by the element ΘH:=ΘS,T(H/F,0)∈Z[G]ΘH:=ΘS,T(H/F,0)∈Z[G]Theta_(H):=Theta_(S,T)(H//F,0)inZ[G]\Theta_{H}:=\Theta_{S, T}(H / F, 0) \in \mathbf{Z}[G]ΘH:=ΘS,T(H/F,0)∈Z[G]. It is therefore not clear how to take the "ratio" of these quantities. Similarly, the role of the ppppp-adic logarithm is played by recG(up)∈I/I2recGâ¡up∈I/I2rec_(G)(u_(p))in I//I^(2)\operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right) \in I / I^{2}recGâ¡(up)∈I/I2 and the role of the ppppp-adic valuation is played by ordG(up)∈Z[G]ordGâ¡up∈Z[G]ord_(G)(u_(p))inZ[G]\operatorname{ord}_{G}\left(u_{\mathfrak{p}}\right) \in \mathbf{Z}[G]ordGâ¡(up)∈Z[G].
For this reason, we introduce in [18] an RRRRR-algebra RLRLR_(L)R_{\mathscr{L}}RL that is generated by an element LLL\mathscr{L}L that plays the role of the analytic LLL\mathscr{L}L-invariant, i.e., the "ratio" between ΘLΘLTheta_(L)\Theta_{L}ΘL and ΘHΘHTheta_(H)\Theta_{H}ΘH. We define
(7.9)RL=R[L]/(ΘHL−ΘL,LI,L2,I2)(7.9)RL=R[L]/ΘHL−ΘL,LI,L2,I2{:(7.9)R_(L)=R[L]//(Theta_(H)L-Theta_(L),LI,L^(2),I^(2)):}\begin{equation*}
R_{\mathscr{L}}=R[\mathscr{L}] /\left(\Theta_{H} \mathscr{L}-\Theta_{L}, \mathscr{L} I, \mathscr{L}^{2}, I^{2}\right) \tag{7.9}
\end{equation*}(7.9)RL=R[L]/(ΘHL−ΘL,LI,L2,I2)
A key nontrivial result is that this ring, in which we have adjoined a ratio LLL\mathscr{L}L between ΘLΘLTheta_(L)\Theta_{L}ΘL and ΘHΘHTheta_(H)\Theta_{H}ΘH, is still large enough to see R/I2R/I2R//I^(2)R / I^{2}R/I2.
Theorem 7.7 ([18, THEOREM 3.4]). The kernel of the structure map R→RLR→RLR rarrR_(L)R \rightarrow R_{\mathscr{L}}R→RL is I2I2I^(2)I^{2}I2.
It follows from this theorem that Gross's Conjecture is equivalent to the equality
(7.10)recG(up)=LordG(up) in RL(7.10)recGâ¡up=LordGâ¡up in RL{:(7.10)rec_(G)(u_(p))=Lord_(G)(u_(p))quad" in "R_(L):}\begin{equation*}
\operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right)=\mathscr{L} \operatorname{ord}_{G}\left(u_{\mathfrak{p}}\right) \quad \text { in } R_{\mathscr{L}} \tag{7.10}
\end{equation*}(7.10)recGâ¡(up)=LordGâ¡(up) in RL
since the right-hand side is by definition LΘH=ΘLLΘH=ΘLLTheta_(H)=Theta_(L)\mathscr{L} \Theta_{H}=\Theta_{L}LΘH=ΘL.
To prove (7.10), we define a generalized Ritter-Weiss module ∇L∇Lgrad_(L)\nabla_{\mathscr{L}}∇L over the ring RLRLR_(L)R_{\mathscr{L}}RL that can be viewed as a gluing of the modules ∇ST(H)∇ST(H)grad_(S)^(T)(H)\nabla_{S}^{T}(H)∇ST(H) and ∇S′T(L)∇S′T(L)grad_(S^('))^(T)(L)\nabla_{S^{\prime}}^{T}(L)∇S′T(L). We show in [18, THEOREM 4.6] that the Fitting ideal Fitt RL(∇L)RL∇LR_(L)(grad_(L))R_{\mathscr{L}}\left(\nabla_{\mathscr{L}}\right)RL(∇L) is generated by the element
recG(up)−LordG(up)∈I/I2recGâ¡up−LordGâ¡up∈I/I2rec_(G)(u_(p))-Lord_(G)(u_(p))in I//I^(2)\operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right)-\mathscr{L} \operatorname{ord}_{G}\left(u_{\mathfrak{p}}\right) \in I / I^{2}recGâ¡(up)−LordGâ¡(up)∈I/I2
(For the sake of accuracy, we remark that in reality we do all of this with (S,T)(S,T)(S,T)(S, T)(S,T) replaced by the pair (Σ,Σ′)Σ,Σ′(Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′) defined in Section 5.3, as in Section 6.)
The vanishing of Fitt RL(∇L)RL∇LR_(L)(grad_(L))R_{\mathscr{L}}\left(\nabla_{\mathscr{L}}\right)RL(∇L) is proven following the methods of Section 6. We interpret surjective homomorphisms from ∇L∇Lgrad_(L)\nabla_{\mathscr{L}}∇L to RLRLR_(L)R_{\mathscr{L}}RL-modules MMMMM in terms of Galois cohomology classes satisfying certain local conditions. We construct a suitable Galois cohomology class valued in a module MMMMM using an explicit construction with group-ring valued Hilbert modular forms and their associated Galois representations. The module MMMMM is shown to be large enough that its Fitting ideal over RLRLR_(L)R_{\mathscr{L}}RL vanishes, whence the same is true for ∇L∇Lgrad_(L)\nabla_{\mathscr{L}}∇L since it has MMMMM as a quotient.
7.5. The method of Darmon-Pozzi-Vonk
We conclude by describing a proof of Conjecture 7.4 in the case that FFFFF is a real quadratic field in the beautiful work of Darmon, Pozzi, and Vonk [14]. Their method is purely ppppp-adic (i.e., "vertical"), rather than involving the introduction of auxiliary primes (i.e., "horizontal"). The strategy follows a rich history of arithmetic formulas proven by exhibiting both sides of an equation as certain Fourier coefficients in an equality of modular forms. For instance, Katz gave an elegant proof of Leopoldt's evaluation of the Kubota-Leopoldt ppppp-adic LLLLL-function at s=1s=1s=1s=1s=1 by exhibiting an equality of ppppp-adic modular forms, one of whose constant terms is the ppppp-adic LLLLL-value and the other is the ppppp-adic logarithm of a unit (see [33, $10.2]). The proof of Darmon-Pozzi-Vonk follows a similar strategy.
Let FFFFF be a real quadratic field, ppppp an odd prime, and HHHHH a narrow ring class field extension of FFFFF (so, in particular, pOFpOFpO_(F)p O_{F}pOF splits completely in HHHHH ). Darmon-Pozzi-Vonk demonstrate an equality of certain classical modular forms of weight 2 on Γ0(p)⊂SL2(Z)Γ0(p)⊂SL2(Z)Gamma_(0)(p)subSL_(2)(Z)\Gamma_{0}(p) \subset \mathrm{SL}_{2}(\mathbf{Z})Γ0(p)⊂SL2(Z) that we denote f1f1f_(1)f_{1}f1 and f2f2f_(2)f_{2}f2.
This first of these forms f1f1f_(1)f_{1}f1 is obtained by considering a Hida family of Hilbert modular cusp forms for FFFFF specializing in weight 1 to a ppppp-stabilized Eisenstein series. The constant term of this weight 1 Eisenstein series vanishes because of the trivial zero of the corresponding ppppp-adic LLLLL-function. Pozzi has described explicitly the Fourier coefficients of the derivative of this family with respect to the weight variables [39]. The key idea of DarmonPozzi-Vonk is to restrict the derivative in the antiparallel direction along the diagonal and take the ordinary projection to obtain a classical modular form of weight 2 for Γ0(p)Γ0(p)Gamma_(0)(p)\Gamma_{0}(p)Γ0(p). The idea of taking the derivative of a family of modular forms at a point of vanishing and applying a "holomorphic projection" operator has its roots in the seminal work of Gross-Zagier [30], and appears more recently in Kudla's program for incoherent Eisenstein series [34].
Pozzi's work relates the ppppp th Fourier coefficient of this diagonal restriction to the ppppp adic logarithm of the Brumer-Stark unit σb(up)σbupsigma_(b)(u_(p))\sigma_{\mathfrak{b}}\left(u_{p}\right)σb(up) for the extension HHHHH. To obtain the desired weight 2 form f1f1f_(1)f_{1}f1 on Γ0(p)Γ0(p)Gamma_(0)(p)\Gamma_{0}(p)Γ0(p), one must take a certain linear combination with the diagonal restrictions of the two ordinary families of Eisenstein series passing through this weight 1 point.
The second form f2f2f_(2)f_{2}f2 is defined as a generating series attached to a certain rigid analytic theta cocycle. These are classes in H1(SL2(Z[1/p]),A∗/Cp∗)H1SL2(Z[1/p]),A∗/Cp∗H^(1)(SL_(2)(Z[1//p]),A^(**)//C_(p)^(**))H^{1}\left(\mathrm{SL}_{2}(\mathbf{Z}[1 / p]), \mathcal{A}^{*} / \mathbf{C}_{p}^{*}\right)H1(SL2(Z[1/p]),A∗/Cp∗), where A∗A∗A^(**)\mathcal{A}^{*}A∗ denotes the group of rigid analytic nonvanishing functions on the ppppp-adic upper half plane. DarmonPozzi-Vonk construct classes in this space explicitly, and study their image under the logarithmic annular residue map
They compute the spectral expansion of the form f2f2f_(2)f_{2}f2 and thereby show that its nonconstant Fourier coefficients are equal to those of f1f1f_(1)f_{1}f1. Meanwhile, the constant coefficient is equal to the ppppp-adic logarithm of up(b)anup(b)anu_(p)(b)^(an)u_{p}(\mathfrak{b})^{\mathrm{an}}up(b)an. The equality of the non-constant coefficients implies that f1=f2f1=f2f_(1)=f_(2)f_{1}=f_{2}f1=f2, and hence that the constant coefficients are equal as well, i.e.,
as desired. It is a tantalizing problem to generalize this strategy to arbitrary totally real fields.
ACKNOWLEDGMENTS
We would like to thank the many mathematicians whose work has been highly influential in the development of the perspective that we have described here. Our work is the continuation of a long line of research connecting LLLLL-functions, modular forms, Galois representations, and Fitting ideals of class groups. In particular, we would like to thank Armand Brumer, David Burns, John Coates, Pierre Charollois, Pierre Colmez, Henri Darmon, Cornelius Greither, Benedict Gross, Masato Kurihara, Cristian Popescu, Alice Pozzi, Kenneth Ribet, Jurgen Ritter, Karl Rubin, Takamichi Sano, Michael Spiess, John Tate, Jan Vonk, Alfred Weiss, and Andrew Wiles.
FUNDING
The first author is supported by a grant from the National Science Foundation, DMS1901939. The second author is supported by DST-SERB grant SB/SJF/2020-21/11, SERB SUPRA grant SPR/2019/000422 and SERB MATRICS grant MTR/2020/000215.
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SAMIT DASGUPTA
Duke University, Department of Mathematics, Campus Box 90320, Durham, NC 277080320, USA, dasgupta@ math.duke.edu
MAHESH KAKDE
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India, maheshkakde@iisc.ac.in
ARITHMETIC AND DYNAMICS ON VARIETIES OF MARKOFF TYPE
ALEXANDER GAMBURD
ABSTRACT
The Markoff equation x2+y2+z2=3xyzx2+y2+z2=3xyzx^(2)+y^(2)+z^(2)=3xyzx^{2}+y^{2}+z^{2}=3 x y zx2+y2+z2=3xyz, which arose in his spectacular thesis in 1879 , is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we discuss Hasse principle, asymptotics of integer points, and, in particular, recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.
Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor.
Hermann Weyl, The Classical Groups, 1939
1. INTRODUCTION
1.1. Andrei Andreevich Markov is one of the towering peaks of the illustrious Saint Petersburg school of number theory, alongside with Chebyshev and Linnik. A singular characteristic of this school is a deep, often subterranean, interaction between arithmetic/combinatorics and probability/dynamics. While Markov is perhaps most widely known today for the chains named after him, it is in the context of his arguably deepest work on the minima of binary quadratic forms and badly approximable numbers 11^(1){ }^{1}1 that the following equation, now bearing his name, was born:
describing a Markoff surface X⊂A3X⊂A3X subA^(3)X \subset \mathbb{A}^{3}X⊂A3. Markoff triples MMM\mathcal{M}M are the solutions of (1.1) with positive integral coordinates. Markoff numbers M⊂NM⊂NMsubN\mathbb{M} \subset \mathbb{N}M⊂N are obtained as coordinates of elements of MMM\mathcal{M}M. The Markoff sequence MsMsM^(s)\mathbb{M}^{s}Ms is the set of largest coordinates of an m∈Mm∈Mm inMm \in \mathcal{M}m∈M counted with multiplicity; the uniqueness conjecture of Frobenius [62] asserts that M=MsM=MsM=M^(s)\mathbb{M}=\mathbb{M}^{s}M=Ms.
All elements of MMM\mathcal{M}M are gotten from the root solution r=(1,1,1)r=(1,1,1)r=(1,1,1)r=(1,1,1)r=(1,1,1) by repeated applications of an element in a set SSSSS, consisting of σ∈Σ3σ∈Σ3sigma inSigma_(3)\sigma \in \Sigma_{3}σ∈Σ3, the permutations of the coordinates of (x1,x2,x3)x1,x2,x3(x_(1),x_(2),x_(3))\left(x_{1}, x_{2}, x_{3}\right)(x1,x2,x3), and of the Vieta involutions R1,R2,R3R1,R2,R3R_(1),R_(2),R_(3)R_{1}, R_{2}, R_{3}R1,R2,R3 of A3A3A^(3)\mathbb{A}^{3}A3, with R1(x1,x2,x3)=(3x2x3−R1x1,x2,x3=3x2x3−R_(1)(x_(1),x_(2),x_(3))=(3x_(2)x_(3)-:}R_{1}\left(x_{1}, x_{2}, x_{3}\right)=\left(3 x_{2} x_{3}-\right.R1(x1,x2,x3)=(3x2x3−x1,x2,x3x1,x2,x3x_(1),x_(2),x_(3)x_{1}, x_{2}, x_{3}x1,x2,x3 ) and R2,R3R2,R3R_(2),R_(3)R_{2}, R_{3}R2,R3 defined similarly. Denoting by ΓΓGamma\GammaΓ the nonlinear group of affine morphisms of A3A3A^(3)\mathbb{A}^{3}A3 generated by SSSSS, the set of Markoff triples MMM\mathcal{M}M can be identified with the orbit of the root rrrrr under the action of ΓΓGamma\GammaΓ, that is to say, M=Γ⋅rM=Γ⋅rM=Gamma*r\mathcal{M}=\Gamma \cdot rM=Γ⋅r, giving rise to the Markoff tree [8]:
1 This work of Markoff and some of the subsequent appearances of his equation in a tremendous variety of different contexts are briefly discussed in Section 2.
The sequence MsMsM^(s)\mathbb{M}^{s}Ms is sparse, as shown by Zagier [147]:
(1.2)∑m∈Msm≤T1∼c(logT)2 as T→∞(c>0)(1.2)∑m∈Msm≤T 1∼c(logâ¡T)2 as T→∞(c>0){:(1.2)sum_({:[m inM^(s)],[m <= T]:})1∼c(log T)^(2)quad" as "T rarr oo(c > 0):}\begin{equation*}
\sum_{\substack{m \in \mathbb{M}^{s} \\ m \leq T}} 1 \sim c(\log T)^{2} \quad \text { as } T \rightarrow \infty(c>0) \tag{1.2}
\end{equation*}(1.2)∑m∈Msm≤T1∼c(logâ¡T)2 as T→∞(c>0)
1.2. The origins of investigations which underlie "the stock and core" of this report date back to August of 2005 and involve a "special problem" pertaining to Markoff numbers; here is Peter Sarnak's recollection [126]: "For me the starting point of this investigation was in 2005 when Michel and Venkatesh asked me about the existence of poorly distributed closed geodesics on the modular surface. It was clear that Markov's constructions of his geodesics using his Markov equation provided what they wanted but they preferred quadratic forms with square free discriminant. This raised the question of sieving in this context of an orbit of a group of (nonlinear) morphisms of affine space. The kind of issues that one quickly faces in attempting to execute such a sieve are questions of the image of the orbit when reduced modqmodqmod q\bmod qmodq and interestingly whether certain graphs associated with these orbits are expander families. 22^(2){ }^{2}2 Gamburd in his thesis had established the expander property in some simpler but similar settings and he and I began a lengthy investigation into this sieving problem in the simpler setting when the group of affine morphisms acts linearly (or what we call now the affine linear sieve)."
The question posed by Michel and Venkatesh arose in the course of their joint work with Einsiedler and Lindenstrauss [58,59] on generalizations of Duke's theorem [57]; formulated in terms of Markoff numbers, it leads to the following:
Conjecture 1. There are infinitely many square-free Markoff numbers.
As detailed in [21], an application of sieve methods in the setting of affine orbits leads to and demands an affirmative answer to the question as to whether Markoff graphs, obtained as a modular reduction of the Markoff tree, 33^(3){ }^{3}3 form a family of expanders. Numerical experiments by de Courcy-Ireland and Lee [55], as well as results detailed in Section 2.5, are compelling in favor of the following superstrong approximation conjecture for Markoff graphs:
Conjecture 2. The family of Markoff graphs X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p) forms a family of expanders.
Before attacking this conjecture, asserting high connectivity of Markoff graphs, one has to confront the question of their connectivity, that is to say, the issue of the strong approximation for Markoff graphs:
FIGURE 1
Markoff graph mod 7. In [54] it is proved that the Markoff graphs are not planar for primes greater than 7.
Conjecture 3. The map πp:M→X∗(p)Ï€p:M→X∗(p)pi_(p):MrarrX^(**)(p)\pi_{p}: \mathcal{M} \rightarrow X^{*}(p)Ï€p:M→X∗(p) is onto, that is to say, Markoff graphs X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p) are connected.
While Conjectures 1 and 2 have withstood our protracted attack over the past 17 years, much progress has been made on parallel questions in the case of affine linear maps. We will return to the recent resolution of Conjecture 3, and resulting progress on diophantine properties of Markoff numbers in Section 1.5.
1.3. Before describing the general setting of Affine Linear Sieve, it is instructive to briefly examine an example which is in many ways parallel to the Markoff situation, namely integral Apollonian packings [63,127]. A theorem of Descartes asserts that x1,x2,x3,x4∈R4x1,x2,x3,x4∈R4x_(1),x_(2),x_(3),x_(4)inR^(4)x_{1}, x_{2}, x_{3}, x_{4} \in \mathbb{R}^{4}x1,x2,x3,x4∈R4 are the curvatures of four mutually tangent circles in the plane if
Given an initial configuration of 4 such circles, we fill in repeatedly the lune regions with the unique circle which is tangent to 3 sides (which is possible by a theorem of Apollonius). In this way we get a packing of the outside circle by circles giving an Apollonian packing. The interesting diophantine feature is that if the initial curvatures are integral then so are the curvatures of the entire packing.
and hence Λ≤OF(Z)Λ≤OF(Z)Lambda <= O_(F)(Z)\Lambda \leq O_{F}(\mathbb{Z})Λ≤OF(Z). The group ΛΛLambda\LambdaΛ is Zariski dense in OFOFO_(F)O_{F}OF, but it is thin in OF(Z)OF(Z)O_(F)(Z)O_{F}(\mathbb{Z})OF(Z). For example, |{γ∈OF(Z):∥γ∥≤T}|∼c1T2γ∈OF(Z):∥γ∥≤T∼c1T2|{gamma inO_(F)(Z):||gamma|| <= T}|∼c_(1)T^(2)\left|\left\{\gamma \in O_{F}(\mathbb{Z}):\|\gamma\| \leq T\right\}\right| \sim c_{1} T^{2}|{γ∈OF(Z):∥γ∥≤T}|∼c1T2 as T→∞T→∞T rarr ooT \rightarrow \inftyT→∞, while |{γ∈Λ:∥γ∥≤T}|∼c1Tδ|{γ∈Λ:∥γ∥≤T}|∼c1Tδ|{gamma in Lambda:||gamma|| <= T}|∼c_(1)T^(delta)|\{\gamma \in \Lambda:\|\gamma\| \leq T\}| \sim c_{1} T^{\delta}|{γ∈Λ:∥γ∥≤T}|∼c1Tδ, where 4δ=1.3…4δ=1.3…^(4)delta=1.3 dots{ }^{4} \delta=1.3 \ldots4δ=1.3… is the Hausdorff dimension of the limit set of ΛΛLambda\LambdaΛ.
The general setting of Affine Linear Sieve, introduced in [20,21], is as follows. For j=1,2,…,kj=1,2,…,kj=1,2,dots,kj=1,2, \ldots, kj=1,2,…,k, let AjAjA_(j)A_{j}Aj be invertible integer coefficient polynomial maps from ZnZnZ^(n)\mathbb{Z}^{n}Zn to ZnZnZ^(n)\mathbb{Z}^{n}Zn (here n≥1n≥1n >= 1n \geq 1n≥1 and the inverses of AjAjA_(j)A_{j}Aj 's are assumed to be of the same type). Let ΛΛLambda\LambdaΛ be the group generated by A1,…,AkA1,…,AkA_(1),dots,A_(k)A_{1}, \ldots, A_{k}A1,…,Ak and let O=ΛbO=ΛbO=Lambda b\mathcal{O}=\Lambda bO=Λb be the orbit of some b∈Znb∈Znb inZ^(n)b \in \mathbb{Z}^{n}b∈Zn under ΛΛLambda\LambdaΛ. Given a polynomial f∈Q[x1,…,xn]f∈Qx1,…,xnf in Q[x_(1),dots,x_(n)]f \in Q\left[x_{1}, \ldots, x_{n}\right]f∈Q[x1,…,xn] which is integral on OOO\mathcal{O}O, the aim is to show that there are many points x∈Ox∈Ox inOx \in \mathcal{O}x∈O at which f(x)f(x)f(x)f(x)f(x) has few or even the least possible number of prime factors, in particular that such points are Zariski dense in the Zariski closure, Zcl(O)Zclâ¡(O)Zcl(O)\operatorname{Zcl}(\mathcal{O})Zclâ¡(O) of OOO\mathcal{O}O. Let O(f,r)O(f,r)O(f,r)\mathcal{O}(f, r)O(f,r) denote the set of x∈Ox∈Ox inOx \in \mathcal{O}x∈O for which f(x)f(x)f(x)f(x)f(x) has at most rrrrr prime factors. As r→∞r→∞r rarr oor \rightarrow \inftyr→∞, the sets O(f,r)O(f,r)O(f,r)\mathcal{O}(f, r)O(f,r) increase and potentially at some point become Zariski dense. Define the saturation number r0(O,f)r0(O,f)r_(0)(O,f)r_{0}(\mathcal{O}, f)r0(O,f) to be the least integer rrrrr such that Zcl(O(f,r))=Zcl(O)Zclâ¡(O(f,r))=Zclâ¡(O)Zcl(O(f,r))=Zcl(O)\operatorname{Zcl}(\mathcal{O}(f, r))=\operatorname{Zcl}(\mathcal{O})Zclâ¡(O(f,r))=Zclâ¡(O). It is by no means obvious that it is finite or even if one should expect it to be so, in general. If it is finite, we say that the pair (O,f)(O,f)(O,f)(\mathcal{O}, f)(O,f) saturates. In the case of linear maps, the theory by now is quite advanced and the basic result pertaining to the finiteness of the saturation number in all cases where it is expected to hold, namely in the case of the Levi factor of G=Zcl(Λ)G=Zclâ¡(Λ)G=Zcl(Lambda)G=\operatorname{Zcl}(\Lambda)G=Zclâ¡(Λ) being semisimple, 5 has been established [123]. Both strong and superstrong approximation, particularly for thin
4 This result can be deduced from the work of Lax and Phillips [93]. A beautiful overview of striking developments pertaining to dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings (and beyond) is contained in Hee Oh's ICM report [114].
5 On the other hand, as detailed in [21, 85, 123], when torus intervenes, the saturation most likely fails. Tori pose particularly difficult problems, in terms of sparsity of elements in an orbit, strong approximation and diophantine properties: see [104] for a discussion of Artin's Conjecture in the context of strong approximation.
groups such as the Apollonian group, are crucial ingredients in executing Brun combinatorial sieve in this setting.
1.4. The strong approximation for SLn(Z)SLn(Z)SL_(n)(Z)\mathrm{SL}_{n}(\mathbb{Z})SLn(Z), asserting that the reduction πqÏ€qpi_(q)\pi_{q}Ï€q modulo qqqqq is onto, is a consequence of the Chinese remainder theorem; its extension to arithmetic groups is far less elementary but well understood [118]. If SSSSS is a finite symmetric generating set of SLn(Z)SLnâ¡(Z)SL_(n)(Z)\operatorname{SL}_{n}(\mathbb{Z})SLnâ¡(Z), strong approximation is equivalent to the assertion that the Cayley graphs E(SLn(Z/qZ),πq(S))ESLn(Z/qZ),Ï€q(S)E(SL_(n)(Z//qZ),pi_(q)(S))\mathcal{E}\left(\mathrm{SL}_{n}(\mathbb{Z} / q \mathbb{Z}), \pi_{q}(S)\right)E(SLn(Z/qZ),Ï€q(S)) are connected. The quantification of this statement, asserting that they are in fact highly-connected, that is to say, form a family of expanders, is what we mean by superstrong approximation. The proof of the expansion property for SL2(Z)SL2(Z)SL_(2)(Z)\mathrm{SL}_{2}(\mathbb{Z})SL2(Z) has its roots in Selberg's celebrated lower bound [131] of 316316(3)/(16)\frac{3}{16}316 for the first eigenvalue of the Laplacian on the hyperbolic surfaces associated with congruence subgroups of SL2(Z)SL2(Z)SL_(2)(Z)\mathrm{SL}_{2}(\mathbb{Z})SL2(Z). The generalization of the expansion property to G(Z)G(Z)G(Z)G(\mathbb{Z})G(Z) where GGGGG is a semisimple matrix group defined over QQQ\mathbb{Q}Q is also known thanks to developments towards the general Ramanujan conjectures that have been established; this expansion property is also referred to as property τÏ„tau\tauÏ„ for congruence subgroups [133].
In my thesis [66], extending the work of Sarnak and Xue [129], [128] for cocompact arithmetic lattices, a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL2(Z)SL2(Z)SL_(2)(Z)\mathrm{SL}_{2}(\mathbb{Z})SL2(Z) was proved; for such subgroups with a high enough Hausdorff dimension of the limit set, a spectral gap property was established. Following the groundbreaking work of Helfgott [77] (which builds crucially on sum-product estimate in FpFpF_(p)\mathbb{F}_{p}Fp due to Bourgain, Katz, and Tao [27]), Bourgain and Gamburd [13] gave a complete answer to Lubotzky's question. The method introduced in [12,13][12,13][12,13][12,13][12,13] and developed in a series of papers [14-19] became known as "Bourgain-Gamburd expansion machine"; thanks to a number of major developments by many people [22,28,35,82,91,115,120,122,124][22,28,35,82,91,115,120,122,124][22,28,35,82,91,115,120,122,124][22,28,35,82,91,115,120,122,124][22,28,35,82,91,115,120,122,124], the general superstrong approximation for thin groups is now known. The state-of-the-art is summarized in Thin groups and superstrong approximation [36] which contains an expanded version of most of the invited lectures from the eponymous MSRI "Hot Topics" workshop, in the surveys by Breuillard [33] and Helfgott [78], and in the book by Tao "Expansion in finite simple groups of Lie type" [140].
1.5. We return to the progress on Conjecture 3 [23-26]. Our first result [25] asserts that there is a very large orbit.
Theorem 1. Fix ε>0ε>0epsi > 0\varepsilon>0ε>0. Then for ppppp large prime, there is a ΓΓGamma\GammaΓ orbit C(p)C(p)C(p)\mathcal{C}(p)C(p) in X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p) for which
(note that |X∗(p)|∼p2X∗(p)∼p2|X^(**)(p)|∼p^(2)\left|X^{*}(p)\right| \sim p^{2}|X∗(p)|∼p2 ), and any ΓΓGamma\GammaΓ orbit D(p)D(p)D(p)\mathscr{D}(p)D(p) satisfies 66^(6)^{6}6
The proof, discussed in section 3, establishes the strong approximation conjecture, unless p2−1p2−1p^(2)-1p^{2}-1p2−1 is a very smooth number. In particular, the set of primes for which the strong approximation conjecture fails is very small.
Theorem 2. Let EEEEE be the set of primes for which the strong approximation conjecture fails. For ε>0ε>0epsi > 0\varepsilon>0ε>0, the number of primes p≤Tp≤Tp <= Tp \leq Tp≤T with p∈Ep∈Ep in Ep \in Ep∈E is at most TεTεT^(epsi)T^{\varepsilon}Tε, for TTTTT large.
Very recently, in a remarkable breakthrough, using geometric techniques involving Hurwitz stacks, degeneration, and some Galois theory, William Chen [45] proved the following result:
Theorem 3. Every ΓΓGamma\GammaΓ orbit D(p)D(p)D(p)\mathscr{D}(p)D(p) has size divisible by ppppp.
Combining Theorems 1 and 3 establishes Conjecture 3 for all sufficiently large primes; in combination with the following result established in [26], namely
Theorem 4. Assume that X∗(Z/pZ)X∗(Z/pZ)X^(**)(Z//pZ)X^{*}(\mathbb{Z} / p \mathbb{Z})X∗(Z/pZ) is connected. Then X∗(Z/pkZ)X∗Z/pkZX^(**)(Z//p^(k)Z)X^{*}\left(\mathbb{Z} / p^{k} \mathbb{Z}\right)X∗(Z/pkZ) is connected for all kkkkk. it yields
Theorem 5. For all sufficiently large primes ppppp, the group ΓΓGamma\GammaΓ acts minimally on X∗(Zp)X∗ZpX^(**)(Z_(p))X^{*}\left(\mathbb{Z}_{p}\right)X∗(Zp).
We remark that Theorem 5 is not true for X∗(R)X∗(R)X^(**)(R)X^{*}(\mathbb{R})X∗(R); cf. section 4.1. While Conjecture 1 remains out of reach, the progress on strong approximation allows us to establish the following result on the diophantine 77^(7){ }^{7}7 properties of Markoff numbers [25]:
Theorem 6. Almost all Markoff numbers are composite, that is,
∑p∈Msp prime ,p≤T1=o(∑m∈Msm≤T1)∑p∈Msp prime ,p≤T 1=o∑m∈Msm≤T 1sum_({:[p inM^(s)],[p" prime "","p <= T]:})1=o(sum_({:[m inM^(s)],[m <= T]:})1)\sum_{\substack{p \in M^{s} \\ p \text { prime }, p \leq T}} 1=o\left(\sum_{\substack{m \in M^{s} \\ m \leq T}} 1\right)∑p∈Msp prime ,p≤T1=o(∑m∈Msm≤T1)
It is worth contrasting this result with the state of knowledge regarding the sequence Hn=2n+bHn=2n+bH_(n)=2^(n)+bH_{n}=2^{n}+bHn=2n+b, which is just a little more sparse than the sequence of Markoff numbers, for which, by Zagier's result (1.2), we have Mn∼AnMn∼AnM_(n)∼A^(sqrtn)M_{n} \sim A^{\sqrt{n}}Mn∼An. Even assuming the generalized Riemann Hypothesis, which allowed Hooley [79] to give a conditional proof of Artin's conjecture (cf. footnote 5), was not sufficient to establish that almost all members of the sequence HnHnH_(n)H_{n}Hn are composite: the conditional proof in [80] necessitated postulating additional "Hypothesis A."
1.6. The methods of proof of Theorems 1, 2, 4 discussed in Section 3 are robust enough to enable handling their extension to more general Markoff-type cubic surfaces, namely
where the real dynamics was studied by Goldman [73], as discussed in Section 4.1; the family of surfaces SA,B,C,D⊂C3SA,B,C,D⊂C3S_(A,B,C,D)subC^(3)S_{A, B, C, D} \subset \mathbb{C}^{3}SA,B,C,D⊂C3 given by
with αij,βj,γ,δαij,βj,γ,δalpha_(ij),beta_(j),gamma,delta\alpha_{i j}, \beta_{j}, \gamma, \deltaαij,βj,γ,δ being integers.
The group ΓYΓYGamma_(Y)\Gamma_{Y}ΓY is again generated by the corresponding Vieta involutions R1,R2,R3R1,R2,R3R_(1),R_(2),R_(3)R_{1}, R_{2}, R_{3}R1,R2,R3. For such a YYYYY and action ΓYΓYGamma_(Y)\Gamma_{Y}ΓY, one must first show that there are only finitely many finite orbits in Y(Q¯)Y(Q¯)Y( bar(Q))Y(\overline{\mathbb{Q}})Y(Q¯), and that these may be determined effectively. The analogue of Conjecture 1 for YYYYY is that for ppppp large, ΓYΓYGamma_(Y)\Gamma_{Y}ΓY has one big orbit on Y(Z/pZ)Y(Z/pZ)Y(Z//pZ)Y(\mathbb{Z} / p \mathbb{Z})Y(Z/pZ) and that the remaining orbits, if there are any, correspond to one of the finite Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ orbits determined above.
In this setting of the more general surfaces YYYYY in (1.8), strong approximation for Y(ZS)YZSY(Z_(S))Y\left(\mathbb{Z}_{S}\right)Y(ZS), where SSSSS is the set of primes dividing α11,α22,α33α11,α22,α33alpha_(11),alpha_(22),alpha_(33)\alpha_{11}, \alpha_{22}, \alpha_{33}α11,α22,α33 (so that ΓYΓYGamma_(Y)\Gamma_{Y}ΓY preserves the SSSSS-integers ZSZSZ_(S)\mathbb{Z}_{S}ZS ), will follow from Conjecture 1 for YYYYY (and the results we can prove towards it, as in Theorem 2) once we have a point of infinite order in Y(ZS)YZSY(Z_(S))Y\left(\mathbb{Z}_{S}\right)Y(ZS). If there is no such point, we can increase SSSSS or replace ZZZ\mathbb{Z}Z by OKOKO_(K)\mathcal{O}_{K}OK, the ring of integers in a number field K/QK/QK//QK / \mathbb{Q}K/Q, to produce such a point and with it strong approximation for Y((OK)S)YOKSY((O_(K))_(S))Y\left(\left(\mathcal{O}_{K}\right)_{S}\right)Y((OK)S).
Vojta's conjectures and the results proven towards them [51,141] assert that cubic and higher-degree affine surfaces typically have few SSSSS-integral points. In the rare cases where these points are Zariski dense, such as tori (e.g., N(x1,x2,x3)=kNx1,x2,x3=kN(x_(1),x_(2),x_(3))=kN\left(x_{1}, x_{2}, x_{3}\right)=kN(x1,x2,x3)=k where NNNNN is the norm form of a cubic extension of QQQ\mathbb{Q}Q ), strong approximation fails. So these Markoff surfaces appear to be rather special affine cubic surfaces not only having a Zariski dense set of integral points, but also a robust strong approximation.
giving an interpretation of the exponent of growth, which for n>3n>3n > 3n>3n>3 is not integral, in terms of the unique parameter for which there exists a certain conformal measure on a projective space.
1.8. The issue of the existence of a single integral solution to (1.9) for general aaaaa and kkkkk, even for n=3n=3n=3n=3n=3, is quite subtle; see [112,130]. In the work of Ghosh and Sarnak [71], the Hasse principle is established to hold for Markoff-type cubic surfaces X(k)X(k)X(k)X(k)X(k) given by (1.6) for almost all kkkkk, but it also fails to hold for infinitely many kkkkk; this work is discussed in Section 6.
1.9. Regrettably, the space/time constraints prevented us from covering cognate results pertaining to arithmetic and dynamics on K3K3K3\mathrm{K} 3K3 surfaces; see [37,65,106,108,109,135][37,65,106,108,109,135][37,65,106,108,109,135][37,65,106,108,109,135][37,65,106,108,109,135] and references therein. The Markoff equation over quadratic imaginary fields is studied in [134]. Potential cryptographic applications of Markoff graphs are discussed in [64].
1.10. To conclude this introduction, let us note that XkXkX_(k)X_{k}Xk is the relative character variety of representations of the fundamental group of a surface of genus 1 with one puncture to SL2SL2SL_(2)\mathrm{SL}_{2}SL2. The action of the mapping class group is that of ΓΓGamma\GammaΓ. More generally, the (affine) relative character variety VkVkV_(k)V_{k}Vk of representation of π1(Σg,n)Ï€1Σg,npi_(1)(Sigma_(g,n))\pi_{1}\left(\Sigma_{g, n}\right)Ï€1(Σg,n), a surface of genus ggggg with nnnnn punctures, into SL2SL2SL_(2)\mathrm{SL}_{2}SL2 is defined over ZZZ\mathbb{Z}Z, and one can study the diophantine properties of Vk(Z)Vk(Z)V_(k)(Z)V_{k}(\mathbb{Z})Vk(Z). In the work of Whang [144-146], it was shown that VkVkV_(k)V_{k}Vk has a projective compactification relative to which VkVkV_(k)V_{k}Vk is "log-Calabi-Yau." According to the conjectures of Vojta, this places VkVkV_(k)V_{k}Vk as being in the same threshold setting as affine cubic surfaces. Moreover, Vk(Z)Vk(Z)V_(k)(Z)V_{k}(\mathbb{Z})Vk(Z) has a full descent in that the mapping class group acts via nonlinear morphisms on Vk(Z)Vk(Z)V_(k)(Z)V_{k}(\mathbb{Z})Vk(Z) with finitely many orbits. These and more general character varieties connected with higher Teichmüller theory offer a rich family of threshold affine varieties for which one can approach the study of integral points.
2. THE UNREASONABLE(?) UBIQUITY OF MARKOFF EQUATION
Markoff equation and numbers appear in a surprising variety of contexts: see, for example, [1] (subtitled Mathematical Journey from Irrational Numbers to Perfect Matchings) and the references therein.
2.1. The Markoff chain. Equation (1.1) was discovered by Markoff in 1879 in his work on badly approximable numbers. As the sentiment 88^(8){ }^{8}8 expressed by Frobenius [62] in 1913 seems to remain true today, we briefly review the context and statement of Markoff's theorem.
Let ααalpha\alphaα be an irrational number. A celebrated theorem of Hurwitz asserts that ααalpha\alphaα admits
infinitely many rational approximations p/qp/qp//qp / qp/q such that |α−pq|<15q2α−pq<15q2|alpha-(p)/(q)| < (1)/(sqrt5q^(2))\left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{5} q^{2}}|α−pq|<15q2, and, moreover, that if ααalpha\alphaα is GL2(Z)GL2(Z)GL_(2)(Z)\mathrm{GL}_{2}(\mathbb{Z})GL2(Z)-equivalent to the Golden Ratio θ1=(1+5)/2θ1=(1+5)/2theta_(1)=(1+sqrt5)//2\theta_{1}=(1+\sqrt{5}) / 2θ1=(1+5)/2, in the sense that α=aθ1+bcθ1+dα=aθ1+bcθ1+dalpha=(atheta_(1)+b)/(ctheta_(1)+d)\alpha=\frac{a \theta_{1}+b}{c \theta_{1}+d}α=aθ1+bcθ1+d for some integers a,b,c,da,b,c,da,b,c,da, b, c, da,b,c,d with ad−bc=±1ad−bc=±1ad-bc=+-1a d-b c= \pm 1ad−bc=±1, the above result is sharp and the constant 1515(1)/(sqrt5)\frac{1}{\sqrt{5}}15 cannot be replaced by any smaller.
Suppose next that ααalpha\alphaα is not GL2(Z)GL2(Z)GL_(2)(Z)\mathrm{GL}_{2}(\mathbb{Z})GL2(Z)-equivalent to θ1θ1theta_(1)\theta_{1}θ1. Then the result of Markoff's doctoral advisors, Korkine and Zolotareff, [88] asserts that ααalpha\alphaα admits infinitely many rational approximations p/qp/qp//qp / qp/q such that |α−pq|<18q2α−pq<18q2|alpha-(p)/(q)| < (1)/(sqrt8q^(2))\left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{8} q^{2}}|α−pq|<18q2, and, moreover, that the constant 1818(1)/(sqrt8)\frac{1}{\sqrt{8}}18 is sharp if and only if ααalpha\alphaα is GL2(Z)GL2(Z)GL_(2)(Z)\mathrm{GL}_{2}(\mathbb{Z})GL2(Z)-equivalent θ2=1+2θ2=1+2theta_(2)=1+sqrt2\theta_{2}=1+\sqrt{2}θ2=1+2.
The general result found by Markoff in his Habilitation and published in 1879 and 1880 in Mathematische Annalen is as follows.
Markoff's Theorem. Let M={1,2,5,13,29,34,89,169,194,…}M={1,2,5,13,29,34,89,169,194,…}M={1,2,5,13,29,34,89,169,194,dots}\mathbb{M}=\{1,2,5,13,29,34,89,169,194, \ldots\}M={1,2,5,13,29,34,89,169,194,…} be the sequence of Markoff numbers. There is a sequence of associated quadratic irrationals θi∈Q(Δi)θi∈QΔitheta_(i)inQ(sqrt(Delta_(i)))\theta_{i} \in \mathbb{Q}\left(\sqrt{\Delta_{i}}\right)θi∈Q(Δi), where Δi=9mi2−4Δi=9mi2−4Delta_(i)=9m_(i)^(2)-4\Delta_{i}=9 m_{i}^{2}-4Δi=9mi2−4 and mimim_(i)m_{i}mi is the iiiii th element of the sequence, with the following property. Let ααalpha\alphaα be a real irrational, not GL2(Z)GL2(Z)GL_(2)(Z)G L_{2}(\mathbb{Z})GL2(Z)-equivalent to any of the numbers θiθitheta_(i)\theta_{i}θi whenever mi<mi<m_(i) <m_{i}<mi<mjmjm_(j)m_{j}mj. Then ααalpha\alphaα admits infinitely many rational approximations p/qp/qp//qp / qp/q with |α−pq|<mjΔjq2α−pq<mjΔjq2|alpha-(p)/(q)| < (m_(j))/(sqrt(Delta_(j))q^(2))\left|\alpha-\frac{p}{q}\right|<\frac{m_{j}}{\sqrt{\Delta_{j}} q^{2}}|α−pq|<mjΔjq2; the constant mj/Δjmj/Δjm_(j)//sqrt(Delta_(j))m_{j} / \sqrt{\Delta_{j}}mj/Δj is sharp if and only if ααalpha\alphaα is GL2(Z)GL2(Z)GL_(2)(Z)G L_{2}(\mathbb{Z})GL2(Z)-equivalent to θhθhtheta_(h)\theta_{h}θh, for some hhhhh such that mh=mjmh=mjm_(h)=m_(j)m_{h}=m_{j}mh=mj.
2.2. Continued fractions and binary quadratic forms. The first paper by Markoff [102] used the theory of continued fractions, while the second memoir [103] was based on the theory of binary indefinite quadratic forms, with the final result stated as a theorem on minima of binary indefinite quadratic forms.
The alternative approach based on indefinite binary quadratic forms was the subject of an important memoir by Frobenius [62] and complete details were finally provided by Remak [121] and much simplified by Cassels [39,40].
2.3. The geometry of Markoff numbers. A third way of looking at the problem, via hyperbolic geometry, was introduced by Gorshkov [74] in his thesis of 1953, but published only in 1977. The connection with hyperbolic geometry was rediscovered, in a somewhat different way, by Cohn [46]. The paper by Caroline Series [132] contains a beautiful exposition of the problem in this context.
2.4. Cohn tree and Nielsen transformations. Cohn is also credited for the interpretation of the problem [47] in the context of primitive words in F2F2F_(2)F_{2}F2, the free group on two generators. Its automorphism group Φ2=Aut(F2)Φ2=Autâ¡F2Phi_(2)=Aut(F_(2))\Phi_{2}=\operatorname{Aut}\left(F_{2}\right)Φ2=Autâ¡(F2) is generated by the following Nielsen transformations: (a,b)P=(b,a),(a,b)σ=(a−1,b),(a,b)U=(a−1,ab)(a,b)P=(b,a),(a,b)σ=a−1,b,(a,b)U=a−1,ab(a,b)^(P)=(b,a),(a,b)^(sigma)=(a^(-1),b),(a,b)^(U)=(a^(-1),ab)(a, b)^{P}=(b, a),(a, b)^{\sigma}=\left(a^{-1}, b\right),(a, b)^{U}=\left(a^{-1}, a b\right)(a,b)P=(b,a),(a,b)σ=(a−1,b),(a,b)U=(a−1,ab). Let V=σUV=σUV=sigma UV=\sigma UV=σU. Then (a,b)V=(a,ab)(a,b)V=(a,ab)(a,b)^(V)=(a,ab)(a, b)^{V}=(a, a b)(a,b)V=(a,ab).
The Cohn tree is a binary tree with root abababa bab, branching to the top with UUUUU and to the bottom with VVVVV,
Markoff numbers are obtained from the Cohn tree by taking a third of the trace of the matrix obtained by substituting the matrices A=(5221)A=5221A=([5,2],[2,1])A=\left(\begin{array}{ll}5 & 2 \\ 2 & 1\end{array}\right)A=(5221) and B=(2111)B=2111B=([2,1],[1,1])B=\left(\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right)B=(2111) in place of a,ba,ba,ba, ba,b and performing the matrix multiplication.
2.5. Nielsen systems and product replacement graphs. Conjecture 3 is a special case of Conjecture QQQQQ made by McCullough and Wanderley [107] in the context of Nielsen systems and product replacement graphs.
Given a group GGGGG, the product replacement graph Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) introduced in [42] in connection with computing in finite groups is defined as follows. The vertices of Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) consist of all kkkkk-tuples of generators (g1,…,gk)g1,…,gk(g_(1),dots,g_(k))\left(g_{1}, \ldots, g_{k}\right)(g1,…,gk) of the group GGGGG. For every (i,j),1≤i(i,j),1≤i(i,j),1 <= i(i, j), 1 \leq i(i,j),1≤i, j≤k,i≠jj≤k,i≠jj <= k,i!=jj \leq k, i \neq jj≤k,i≠j, there is an edge corresponding to transformations Li,j±Li,j±L_(i,j)^(+-)L_{i, j}^{ \pm}Li,j±and Ri,j±Ri,j±R_(i,j)^(+-)R_{i, j}^{ \pm}Ri,j±, where Ri,j±:(g1,…,gi,…,gk)→(g1,…,gi⋅gj±1,…,gk)Ri,j±:g1,…,gi,…,gk→g1,…,giâ‹…gj±1,…,gkR_(i,j)^(+-):(g_(1),dots,g_(i),dots,g_(k))rarr(g_(1),dots,g_(i)*g_(j)^(+-1),dots,g_(k))R_{i, j}^{ \pm}:\left(g_{1}, \ldots, g_{i}, \ldots, g_{k}\right) \rightarrow\left(g_{1}, \ldots, g_{i} \cdot g_{j}^{ \pm 1}, \ldots, g_{k}\right)Ri,j±:(g1,…,gi,…,gk)→(g1,…,giâ‹…gj±1,…,gk) and Li,j±Li,j±L_(i,j)^(+-)L_{i, j}^{ \pm}Li,j±defined similarly. The graphs Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) are regular, of degree 4k(k−1)4k(k−1)4k(k-1)4 k(k-1)4k(k−1), possibly with loops and multiple edges. The connectivity of Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) has been the subject of intensive recent investigations; for G=SL2(p)G=SL2(p)G=SL_(2)(p)G=\mathrm{SL}_{2}(p)G=SL2(p) and k≥3k≥3k >= 3k \geq 3k≥3, it was established by Gilman in [72].
In the case of the free group FkFkF_(k)F_{k}Fk, the moves Li,j±Li,j±L_(i,j)^(+-)L_{i, j}^{ \pm}Li,j±and Ri,j±Ri,j±R_(i,j)^(+-)R_{i, j}^{ \pm}Ri,j±defined above correspond to Nielsen moves on Γk(Fk)ΓkFkGamma_(k)(F_(k))\Gamma_{k}\left(F_{k}\right)Γk(Fk). For every group GGGGG, the set Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) can be identified with E=Epi(Fk,G)E=Epiâ¡Fk,GE=Epi(F_(k),G)E=\operatorname{Epi}\left(F_{k}, G\right)E=Epiâ¡(Fk,G), the set of epimorphisms from FkFkF_(k)F_{k}Fk onto GGGGG, and the group A=Aut(Fk)A=Autâ¡FkA=Aut(F_(k))A=\operatorname{Aut}\left(F_{k}\right)A=Autâ¡(Fk) acts on EEEEE in the following way: if α∈Aα∈Aalpha in A\alpha \in Aα∈A and φ∈E,α(φ)=φ⋅α−1φ∈E,α(φ)=φ⋅α−1varphi in E,alpha(varphi)=varphi*alpha^(-1)\varphi \in E, \alpha(\varphi)=\varphi \cdot \alpha^{-1}φ∈E,α(φ)=φ⋅α−1. A long-standing problem is whether Aut(Fk)Autâ¡FkAut(F_(k))\operatorname{Aut}\left(F_{k}\right)Autâ¡(Fk) has property (T) for k≥4k≥4k >= 4k \geq 4k≥4; in [100] Lubotzky and Pak observed that a positive answer to this problem implies the expansion of Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) for all GGGGG and proved that Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) are expanders when GGGGG is nilpotent of class lllll and both kkkkk and lllll are fixed. Property (T) for Aut(Fk)Autâ¡FkAut(F_(k))\operatorname{Aut}\left(F_{k}\right)Autâ¡(Fk) for k≥5k≥5k >= 5k \geq 5k≥5 was recently established in [84]. 99^(9){ }^{9}9 Note that Aut(F2)Autâ¡F2Aut(F_(2))\operatorname{Aut}\left(F_{2}\right)Autâ¡(F2) and Aut(F3)Autâ¡F3Aut(F_(3))\operatorname{Aut}\left(F_{3}\right)Autâ¡(F3) do not satisfy property (T), while the problem is still open for k=4k=4k=4k=4k=4.
In a joint work with Pak [69], we established a connection between the expansion coefficient of the product replacement graph Γk(G)Γk(G)Gamma_(k)(G)\Gamma_{k}(G)Γk(G) and the minimal expansion coefficient of a Cayley graph of GGGGG with kkkkk generators, and, in particular, proved that for k>3k>3k > 3k>3k>3 the product
9 The proof stems from the groundbreaking observation by Ozawa [116] that GGGGG satisfies Kazhdan's property (T) if there exist λ>0λ>0lambda > 0\lambda>0λ>0 and finitely many elements ξiξixi_(i)\xi_{i}ξi of R[G]R[G]R[G]\mathbb{R}[G]R[G] such that Δ2−λΔ=∑iξi∗ξiΔ2−λΔ=∑i ξi∗ξiDelta^(2)-lambda Delta=sum_(i)xi_(i)^(**)xi_(i)\Delta^{2}-\lambda \Delta=\sum_{i} \xi_{i}^{*} \xi_{i}Δ2−λΔ=∑iξi∗ξi where ΔΔDelta\DeltaΔ is the Laplacian of the finite symmetric generating set of GGGGG.
replacement graphs Γk(SL(2,p))Γk(SLâ¡(2,p))Gamma_(k)(SL(2,p))\Gamma_{k}(\operatorname{SL}(2, p))Γk(SLâ¡(2,p)) form an expander family under assumption of strong uniform expansion of SL (2,p)(2,p)(2,p)(2, p)(2,p) on kkkkk generators. In a joint work with Breuillard [34], combining the "expansion machine" [13] with the uniform Tits Alternative 1010^(10){ }^{10}10 established by Breuillard [32], we proved that Cayley graphs of SL(2,p)SLâ¡(2,p)SL(2,p)\operatorname{SL}(2, p)SLâ¡(2,p) are strongly uniformly expanding for infinitely many primes of density one. Consequently, the following form of nonlinear superstrong approximation is obtained:
Theorem 7. Let k>3k>3k > 3k>3k>3. The family of product replacement graphs {Γk(SL(2,pn))}nΓkSLâ¡2,pnn{Gamma_(k)(SL(2,p_(n)))}_(n)\left\{\Gamma_{k}\left(\operatorname{SL}\left(2, p_{n}\right)\right)\right\}_{n}{Γk(SLâ¡(2,pn))}n forms a family of expanders for infinitely many primes pnpnp_(n)p_{n}pn of density one.
As detailed in [107], the situation is different for the product replacement graph of SL(2,Fp)SLâ¡2,FpSL(2,F_(p))\operatorname{SL}\left(2, \mathbb{F}_{p}\right)SLâ¡(2,Fp) on 2 generators, due to Fricke identity for 2×22×22xx22 \times 22×2 matrices MMMMM and NNNNN :
Letting x1=tr(M),x2=tr(N),x3=tr(MN)x1=trâ¡(M),x2=trâ¡(N),x3=trâ¡(MN)x_(1)=tr(M),x_(2)=tr(N),x_(3)=tr(MN)x_{1}=\operatorname{tr}(M), x_{2}=\operatorname{tr}(N), x_{3}=\operatorname{tr}(M N)x1=trâ¡(M),x2=trâ¡(N),x3=trâ¡(MN), the QQQQQ conjecture 1111^(11){ }^{11}11 in [107] amounts to the assertion of the strong approximation for the surfaces
and k=tr([M,N])+2k=trâ¡([M,N])+2k=tr([M,N])+2k=\operatorname{tr}([M, N])+2k=trâ¡([M,N])+2, with Markoff surface 1212^(12){ }^{12}12 being the special case corresponding to tr([M,N])=−2trâ¡([M,N])=−2tr([M,N])=-2\operatorname{tr}([M, N])=-2trâ¡([M,N])=−2.
3. STRONG APPROXIMATION
We give a brief overview of the methods and tools used in the proof of Theorems 1 and 2 and some comments about their extensions to the setting of more general surfaces of Markoff type. Theorem 1, in the weaker form that |C(p)|∼|X∗(p)||C(p)|∼X∗(p)|C(p)|∼|X^(**)(p)||\mathcal{C}(p)| \sim\left|X^{*}(p)\right||C(p)|∼|X∗(p)| as p→∞p→∞p rarr oop \rightarrow \inftyp→∞, can be viewed as the finite field analogue of [73] where it is shown that the action of ΓΓGamma\GammaΓ on the compact real components of the relative character variety of the mapping class group of the once punctured torus is ergodic. As in [73] our proof makes use of the rotations τij∘RjÏ„ij∘Rjtau_(ij)@R_(j)\tau_{i j} \circ R_{j}Ï„ij∘Rj, i≠ji≠ji!=ji \neq ji≠j, where τijÏ„ijtau_(ij)\tau_{i j}Ï„ij permutes xixix_(i)x_{i}xi and xjxjx_(j)x_{j}xj. These preserve the conic sections gotten by intersecting X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p) with the planes yk=xkyk=xky_(k)=x_(k)y_{k}=x_{k}yk=xk ( kkkkk different from iiiii and jjjjj ). If τij∘RjÏ„ij∘Rjtau_(ij)@R_(j)\tau_{i j} \circ R_{j}Ï„ij∘Rj has order t1t1t_(1)t_{1}t1 (here t1∣p(p−1)(p+1))t1∣p(p−1)(p+1){:t_(1)∣p(p-1)(p+1))\left.t_{1} \mid p(p-1)(p+1)\right)t1∣p(p−1)(p+1)), then xxxxx and these t1t1t_(1)t_{1}t1 points of the conic section are connected (i.e., are in the same ΓΓGamma\GammaΓ orbit). If t1t1t_(1)t_{1}t1 is maximal (i.e., is p,p−1p,p−1p,p-1p, p-1p,p−1, or p+1p+1p+1p+1p+1 ), then this entire conic section is connected and such conic sections in different planes which intersect are also connected. This leads to a large component which we denote by と(p)ã¨(p)ã¨(p)ã¨(p)ã¨ã¨(p).
10 This states that if the subgroup of GLd(K)GLd(K)GL_(d)(K)\mathrm{GL}_{d}(K)GLd(K) (where KKKKK is an algebraic number field) generated by FFFFF is not virtually solvable, then there is some N∈NN∈NN inNN \in \mathbb{N}N∈N, depending only on ddddd, such that (F∪F−1∪{1})NF∪F−1∪{1}N(F uuF^(-1)uu{1})^(N)\left(F \cup F^{-1} \cup\{1\}\right)^{N}(F∪F−1∪{1})N contains two elements that generate a nonabelian free group.
11 See the paper of Will Chen [45] for the discussion of the relation between this conjecture and the connectivity properties of the moduli spaces of elliptic curves with G=SL(2,p)G=SLâ¡(2,p)G=SL(2,p)G=\operatorname{SL}(2, p)G=SLâ¡(2,p) structures.
12 Note that the congruence x2+y2+z2≡xyz(mod3)x2+y2+z2≡xyz(mod3)x^(2)+y^(2)+z^(2)-=xyz(mod3)x^{2}+y^{2}+z^{2} \equiv x y z(\bmod 3)x2+y2+z2≡xyz(mod3) has no nontrivial solutions.
If our starting rotation has order t1t1t_(1)t_{1}t1 which is not maximal, then the idea is to ensure that among the t1t1t_(1)t_{1}t1 points to which it is connected, at least one has a corresponding rotation of order t2>t1t2>t1t_(2) > t_(1)t_{2}>t_{1}t2>t1, and then to repeat. To ensure that one can progress in this way, a critical equation over FpFpF_(p)\mathbb{F}_{p}Fp intervenes:
(3.1){x+bx=y+1y,b≠1x∈H1,y∈H2 with H1,H2 subgroups of Fp∗( or Fp2∗)(3.1)x+bx=y+1y,b≠1x∈H1,y∈H2 with H1,H2 subgroups of Fp∗ or Fp2∗{:(3.1){[x+(b)/(x)=y+(1)/(y)","quad b!=1],[x inH_(1)","y inH_(2)" with "H_(1)","H_(2)" subgroups of "F_(p)^(**)(" or "F_(p^(2))^(**))]:}:}\left\{\begin{array}{l}
x+\frac{b}{x}=y+\frac{1}{y}, \quad b \neq 1 \tag{3.1}\\
x \in H_{1}, y \in H_{2} \text { with } H_{1}, H_{2} \text { subgroups of } \mathbb{F}_{p}^{*}\left(\text { or } \mathbb{F}_{p^{2}}^{*}\right)
\end{array}\right.(3.1){x+bx=y+1y,b≠1x∈H1,y∈H2 with H1,H2 subgroups of Fp∗( or Fp2∗)
If t1=|H1|≥p1/2+δt1=H1≥p1/2+δt_(1)=|H_(1)| >= p^(1//2+delta)t_{1}=\left|H_{1}\right| \geq p^{1 / 2+\delta}t1=|H1|≥p1/2+δ (with δδdelta\deltaδ small and fixed), one can apply the proven Riemann Hypothesis for curves over finite fields [142] to count the number of solutions to (3.1). Together with a simple inclusion/exclusion argument, this shows that one of the t1t1t_(1)t_{1}t1 points connected to our starting xxxxx has a corresponding maximal rotation and hence xxxxx is connected to ℓ(p)â„“(p)â„“(p)\ell(p)â„“(p).
If |H1|≤p1/2+δH1≤p1/2+δ|H_(1)| <= p^(1//2+delta)\left|H_{1}\right| \leq p^{1 / 2+\delta}|H1|≤p1/2+δ then RHRHRH\mathrm{RH}RH for these curves is of little use (their genus is too large), and we have to proceed using other methods. We assume that |H1|≥|H2|H1≥H2|H_(1)| >= |H_(2)|\left|H_{1}\right| \geq\left|H_{2}\right||H1|≥|H2| so that the trivial upper bound for the number of solutions to (3.1) is 2|H2|2H22|H_(2)|2\left|H_{2}\right|2|H2|. What we need is a power saving in this upper bound in the case that |H2|H2|H_(2)|\left|H_{2}\right||H2| is close to |H1|H1|H_(1)|\left|H_{1}\right||H1|, that is, a bound of the form Cτ|H1|τCÏ„H1Ï„C_(tau)|H_(1)|^(tau)C_{\tau}\left|H_{1}\right|^{\tau}CÏ„|H1|Ï„, with τ<1,Cτ<∞Ï„<1,CÏ„<∞tau < 1,C_(tau) < oo\tau<1, C_{\tau}<\inftyÏ„<1,CÏ„<∞ (both fixed).
In the prime modulus case, there are several ways to proceed. The first and second methods are related to "elementary" proofs of the Riemann Hypothesis for curves. One can use auxiliary polynomials as in Stepanov's proof [137] of the Riemann Hypothesis for curves to give the desired power saving with an explicit τÏ„tau\tauÏ„ (cf. [76] which deals with x+y=1x+y=1x+y=1x+y=1x+y=1 and |H1|=|H2|H1=H2|H_(1)|=|H_(2)|\left|H_{1}\right|=\left|H_{2}\right||H1|=|H2| in (3.1)). The second method, giving the best upper bound, namely 20max{(|H1|.|H2|)1/3,|H1|.|H2|p}20maxH1.H21/3,H1.H2p20 max{(|H_(1)|.|H_(2)|)^(1//3),(|H_(1)|.|H_(2)|)/(p)}20 \max \left\{\left(\left|H_{1}\right| .\left|H_{2}\right|\right)^{1 / 3}, \frac{\left|H_{1}\right| .\left|H_{2}\right|}{p}\right\}20max{(|H1|.|H2|)1/3,|H1|.|H2|p}, is due to Corvaja and Zannier [53]. It uses their method for estimating the greatest common divisor of u−1u−1u-1u-1u−1 and v−1v−1v-1v-1v−1 in terms of the degrees of uuuuu and vvvvv and their supports, as well as (hyper) Wronskians.
Given ε>0,r>1ε>0,r>1epsi > 0,r > 1\varepsilon>0, r>1ε>0,r>1, there is δ>0δ>0delta > 0\delta>0δ>0 such that if A⊂P1(Fp)A⊂P1FpA subP^(1)(F_(p))A \subset P^{1}\left(\mathbb{F}_{p}\right)A⊂P1(Fp) and L⊂FpL⊂FpL subF_(p)L \subset \mathbb{F}_{p}L⊂Fp satisfy
(3.4)|{(x,y,t)∈A×A×L;y=τΦ(t)(x)}|<|A|1−δ|L|(3.4)(x,y,t)∈A×A×L;y=τΦ(t)(x)<|A|1−δ|L|{:(3.4)|{(x,y,t)in A xx A xx L;y=tau_(Phi(t))(x)}| < |A|^(1-delta)|L|:}\begin{equation*}
\left|\left\{(x, y, t) \in A \times A \times L ; y=\tau_{\Phi(t)}(x)\right\}\right|<|A|^{1-\delta}|L| \tag{3.4}
\end{equation*}(3.4)|{(x,y,t)∈A×A×L;y=τΦ(t)(x)}|<|A|1−δ|L|
where for g=(abcd),τg(x)=ax+bcx+dg=abcd,Ï„g(x)=ax+bcx+dg=([a,b],[c,d]),tau_(g)(x)=(ax+b)/(cx+d)g=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), \tau_{g}(x)=\frac{a x+b}{c x+d}g=(abcd),Ï„g(x)=ax+bcx+d.
While producing poor exponents τÏ„tau\tauÏ„, this method is robust and works in the generality that the superstrong approximation for SL2(Z/qZ)SL2(Z/qZ)SL_(2)(Z//qZ)\mathrm{SL}_{2}(\mathbb{Z} / q \mathbb{Z})SL2(Z/qZ) has been established; in particular,
the analogue of Theorem 8 for Z/pnZZ/pnZZ//p^(n)Z\mathbb{Z} / p^{n} \mathbb{Z}Z/pnZ, which follows from expansion in SL2(Z/pnZ)SL2Z/pnZSL_(2)(Z//p^(n)Z)\mathrm{SL}_{2}\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)SL2(Z/pnZ), established 1313^(13){ }^{13}13 in [16], plays crucial role in the proof of Theorem 4 in [26].
The above leads to a proof of part 1 of Theorem 1. To continue, one needs to deal with t1t1t_(1)t_{1}t1 which is very small (here |H1|=t1H1=t1|H_(1)|=t_(1)\left|H_{1}\right|=t_{1}|H1|=t1 which divides p2−1p2−1p^(2)-1p^{2}-1p2−1 ).
To handle these, we lift to characteristic zero and examine the finite orbits of ΓΓGamma\GammaΓ in X(Q¯)X(Q¯)X( bar(Q))X(\overline{\mathbb{Q}})X(Q¯). In fact, by the Chebotarev Density Theorem, a necessary condition for Conjecture 3 to hold is that there are no such orbits other than {0}{0}{0}\{0\}{0}. Again using the rotations in the conic sections by planes, one finds that any such finite orbit must be among the solutions with tjtjt_(j)t_{j}tj 's roots of unity to
For this particular surface XXXXX, one can show using the inequality between the geometric and arithmetic means, that (3.5) has no nontrivial solutions for complex numbers with |tj|=1tj=1|t_(j)|=1\left|t_{j}\right|=1|tj|=1. For the more general surfaces Xk,SA,B,C,DXk,SA,B,C,DX_(k),S_(A,B,C,D)X_{k}, S_{A, B, C, D}Xk,SA,B,C,D, and those in (1.8), there is a variety of solutions with |tj|=1tj=1|t_(j)|=1\left|t_{j}\right|=1|tj|=1. However, Lang's GmGmG_(m)\mathbb{G}_{m}Gm Conjecture which is established effectively (see [2,92][2,92][2,92][2,92][2,92] ) yields that there are only finitely many solutions to these equations in roots of unity. This allows for an explicit determination of the finite orbits of ΓYΓYGamma_(Y)\Gamma_{Y}ΓY in Y(Q¯)Y(Q¯)Y( bar(Q))Y(\overline{\mathbb{Q}})Y(Q¯) (as noted earlier for the cubic surfaces SA,B,C,DSA,B,C,DS_(A,B,C,D)S_{A, B, C, D}SA,B,C,D, the long list of these orbits [96] correspond to the algebraic Painleve VI's). This Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ analysis leads to part 2 of Theorem 1 and, combined with the discussion above, it yields a proof of Conjecture 3, at least if p2−1p2−1p^(2)-1p^{2}-1p2−1 is not very smooth. To prove Theorem 2, we need to show that there are very few primes for which the above arguments fail. This is done by extending the arguments and results in [43] and [44] concerning points (x,y)(x,y)(x,y)(x, y)(x,y) on irreducible curves over FpFpF_(p)\mathbb{F}_{p}Fp for which ord (x)+ord(y)(x)+ordâ¡(y)(x)+ord(y)(x)+\operatorname{ord}(y)(x)+ordâ¡(y) is small (here ord (x)(x)(x)(x)(x) is the order of xxxxx in Fp∗Fp∗F_(p)^(**)\mathbb{F}_{p}^{*}Fp∗ ).
The proof of Theorem 6 in the stronger form that all Markoff numbers are highly composite, that is, for every v≥1v≥1v >= 1v \geq 1v≥1, as T→∞T→∞T rarr ooT \rightarrow \inftyT→∞,
∑m∈Ms,m≤Tm has at most v distinct prime factors 1=o(∑m∈Msm≤T1)∑m∈Ms,m≤Tm has at most v distinct prime factors  1=o∑m∈Msm≤T 1sum_({:[m inM^(s)","m <= T],[m" has at most "],[v" distinct prime factors "]:})1=o(sum_({:[m inM^(s)],[m <= T]:})1)\sum_{\substack{m \in M^{s}, m \leq T \\ m \text { has at most } \\ v \text { distinct prime factors }}} 1=o\left(\sum_{\substack{m \in M^{s} \\ m \leq T}} 1\right)∑m∈Ms,m≤Tm has at most v distinct prime factors 1=o(∑m∈Msm≤T1)
makes use of counting points on X∗(Z)X∗(Z)X^(**)(Z)X^{*}(\mathbb{Z})X∗(Z) of height at most TTTTT and, in particular, Mirzakhani's orbit equidistribution [111], as well as the transitivity properties of ΓΓGamma\GammaΓ on X∗(q)X∗(q)X^(**)(q)X^{*}(q)X∗(q) for qqqqq a product of suitable primes ppppp. The latter are provided by the results of Meiri and Puder [110]. For p≡1(4)p≡1(4)p-=1(4)p \equiv 1(4)p≡1(4) for which the induced permutation action of ΓΓGamma\GammaΓ on X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p) is transitive, they show that the resulting permutation group is essentially the full symmetric or alternating group on X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p). Applying Goursat's (disjointness) Lemma leads to the ΓΓGamma\GammaΓ-action on X∗(p1p2⋯pk)X∗p1p2⋯pkX^(**)(p_(1)p_(2)cdotsp_(k))X^{*}\left(p_{1} p_{2} \cdots p_{k}\right)X∗(p1p2⋯pk) being transitive for any such primes p1<p2<⋯<pkp1<p2<⋯<pkp_(1) < p_(2) < cdots < p_(k)p_{1}<p_{2}<\cdots<p_{k}p1<p2<⋯<pk.
FIGURE 3
Level set κ=−2.1κ=−2.1kappa=-2.1\kappa=-2.1κ=−2.1.
FIGURE 4
Level set κ=−1.9κ=−1.9kappa=-1.9\kappa=-1.9κ=−1.9.
4. REAL DYNAMICS ON SURFACES OF MARKOFF TYPE
4.1. In this section we discuss the work of Goldman [73] pertaining to modular group action on real SL(2)-characters of a one-holed torus. The fundamental group πÏ€pi\piÏ€ of the one-holed torus is the free group of rank two. The mapping class group of the 1-holed torus is isomorphic to the outer automorphism group Out(π)≅GL(2,Z)Outâ¡(Ï€)≅GLâ¡(2,Z)Out(pi)~=GL(2,Z)\operatorname{Out}(\pi) \cong \operatorname{GL}(2, \mathbb{Z})Outâ¡(Ï€)≅GLâ¡(2,Z) of πÏ€pi\piÏ€ and acts on the moduli space of equivalence classes of SL(2,C)SLâ¡(2,C)SL(2,C)\operatorname{SL}(2, \mathbb{C})SLâ¡(2,C)-representations of πÏ€pi\piÏ€; this moduli space identifies naturally with affine 3 -space C3C3C^(3)\mathbb{C}^{3}C3, using the traces of two generators of πÏ€pi\piÏ€ and of their product as coordinates. In these coordinates, the trace of the commutator of the two generators (representing the boundary curve of the torus) is given by κ(x,y,z)=x2+y2+κ(x,y,z)=x2+y2+kappa(x,y,z)=x^(2)+y^(2)+\kappa(x, y, z)=x^{2}+y^{2}+κ(x,y,z)=x2+y2+z2−xyz−2z2−xyz−2z^(2)-xyz-2z^{2}-x y z-2z2−xyz−2, which is preserved under the action of Out(π)Outâ¡(Ï€)Out(pi)\operatorname{Out}(\pi)Outâ¡(Ï€), and the action of Out(π)Outâ¡(Ï€)Out(pi)\operatorname{Out}(\pi)Outâ¡(Ï€) on C3C3C^(3)\mathbb{C}^{3}C3 is commensurable with the action of the group ΓΓGamma\GammaΓ of polynomial automorphisms of C3C3C^(3)\mathbb{C}^{3}C3 which preserve κκkappa\kappaκ. Figures 3-8 show level sets of κκkappa\kappaκ.
4.2. The main objective of [38] is the dynamical description of elements of the mapping class group of the four-punctured sphere acting on two-dimensional slices of its
An element f∈Γf∈Γf in Gammaf \in \Gammaf∈Γ is called hyperbolic if it corresponds to a pseudo-Anosov automorphism in the mapping class group, or, equivalently, if it is not conjugated to the product of
Problem 2. Is β(n)β(n)beta(n)\beta(n)β(n) irrational?
In [68] a complete answer to Problem 1 was obtained by extending Baragar's exponential rate of growth estimate to a true asymptotic formula. 1515^(15){ }^{15}15
When k>0k>0k > 0k>0k>0, there are certain exceptional families of solutions to (5.1) that have a different quality of growth and, for fixed k,a,nk,a,nk,a,nk, a, nk,a,n, we write &&&\&& for the set of exceptional tuples. We obtain the following theorem for the asymptotic number of Markoff-Hurwitz tuples:
Theorem 9. For each (n,a,k)(n,a,k)(n,a,k)(n, a, k)(n,a,k) with V(Z)−EV(Z)−EV(Z)-EV(\mathbb{Z})-\mathcal{E}V(Z)−E infinite, there is a positive constant c=c=c=c=c=c(n,a,k)c(n,a,k)c(n,a,k)c(n, a, k)c(n,a,k) such that
on ordered tuples of real numbers. Above, ∙^∙^widehat(∙)\widehat{\bullet}∙^ denotes omission. If sufficiently many of the ziziz_(i)z_{i}zi are large, the move λjλjlambda_(j)\lambda_{j}λj can be approximated by
any element of ΓΓGamma\GammaΓ maps ordered tuples in R≥0nR≥0nR_( >= 0)^(n)\mathbf{R}_{\geq 0}^{n}R≥0n into HHH\mathscr{H}H. Therefore the study of orbits of ΓΓGamma\GammaΓ and Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ on ordered tuples boils down to the study of orbits in HHH\mathscr{H}H. We can use the basis
for the subspace spanned by HHH\mathscr{H}H. This basis clarifies the action of Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′.
When n=3n=3n=3n=3n=3, the linear map σ:H→Hσ:H→Hsigma:HrarrH\sigma: \mathscr{H} \rightarrow \mathscr{H}σ:H→H defined by
(5.7)σ(a,b,a+b)=order(b−a,a,b)(5.7)σ(a,b,a+b)=orderâ¡(b−a,a,b){:(5.7)sigma(a","b","a+b)=order(b-a","a","b):}\begin{equation*}
\sigma(a, b, a+b)=\operatorname{order}(b-a, a, b) \tag{5.7}
\end{equation*}(5.7)σ(a,b,a+b)=orderâ¡(b−a,a,b)
where order puts a tuple in ascending order from left to right, is such that for j=1,2j=1,2j=1,2j=1,2j=1,2 we have σγj.y=yσγj.y=ysigmagamma_(j).y=y\sigma \gamma_{j} . y=yσγj.y=y for all y∈Hy∈Hy inHy \in \mathscr{H}y∈H. Repeatedly applying the map σσsigma\sigmaσ to a triple (a,b,a+b)(a,b,a+b)(a,b,a+b)(a, b, a+b)(a,b,a+b) with a≤b∈Za≤b∈Za <= b inZa \leq b \in \mathbb{Z}a≤b∈Z performs the Euclidean algorithm on a,ba,ba,ba, ba,b. However, one application of σσsigma\sigmaσ corresponds in general to less than one step of the algorithm. Replacing ΓΓGamma\GammaΓ with Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ corresponds to speeding this up so one whole step of the Euclidean algorithm corresponds to one semigroup generator. As for counting, the orbit of (0,1,1)(0,1,1)(0,1,1)(0,1,1)(0,1,1) under ΓΓGamma\GammaΓ is precisely those (a,b,a+b)(a,b,a+b)(a,b,a+b)(a, b, a+b)(a,b,a+b) with (a,b)=1(a,b)=1(a,b)=1(a, b)=1(a,b)=1 and thus can be counted by elementary methods.
When n=3n=3n=3n=3n=3, the semigroup Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ is generated by
with respect to the basis {e1,e2}e1,e2{e_(1),e_(2)}\left\{e_{1}, e_{2}\right\}{e1,e2}. These generators are classically connected with continued fractions by the formulae
When n=4n=4n=4n=4n=4, the semigroup elements map Δ=H/R+Δ=H/R+Delta=H//R+\Delta=\mathbb{H} / \mathbb{R}+Δ=H/R+ into a strictly smaller subset. After iteration, this leads to more and more empty space (see also Figure 16). This does not occur when n=3n=3n=3n=3n=3, as one can also see from the picture: the action of the group elements γ2γ2gamma_(2)\gamma_{2}γ2 and γ3γ3gamma_(3)\gamma_{3}γ3 on the vertical coordinate axis is a copy of the n=3n=3n=3n=3n=3 dynamics
When n=4n=4n=4n=4n=4, the semigroup ΓΓGamma\GammaΓ acts in the basis given by the eieie_(i)e_{i}ei as
This semigroup appears naturally in different areas of mathematics. In most situations that this semigroup appears, as is the case in [68], the dynamics of the projective linear action of ΓΓGamma\GammaΓ on R+3/R+R+3/R+R_(+)^(3)//R_(+)\mathbb{R}_{+}^{3} / \mathbb{R}_{+}R+3/R+becomes relevant. Up to the minor modification of possibly multiplying the generators on the left or right by permutation matrices, the iterated function system given by the projective linear action of ΓΓGamma\GammaΓ on R+3/R+R+3/R+R_(+)^(3)//R_(+)\mathbb{R}_{+}^{3} / \mathbf{R}_{+}R+3/R+has a fractal attracting set that is known as the Rauzy gasket [95].
So the semigroups ΓΓGamma\GammaΓ and Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ are natural extensions of the Euclidean algorithm and continued fractions semigroup to higher dimensions. Writing Δ=H/R+Δ=H/R+Delta=H//R_(+)\Delta=\mathscr{H} / \mathbb{R}_{+}Δ=H/R+, we can view ΔΔDelta\DeltaΔ as a subset of Rn−2Rn−2R^(n-2)\mathbb{R}^{n-2}Rn−2. The key distinction that appears when n≥4n≥4n >= 4n \geq 4n≥4 is that
and so the induced dynamics on H/R+H/R+H//R_(+)\mathscr{H} / \mathbb{R}_{+}H/R+has "holes" as we illustrate in Figure 15.
Structure of the proof and the difficulties that arise. Here we highlight some of the main difficulties that must be overcome during the proof of Theorem 9. It is illuminating to recall the methods used by Lalley 1616^(16){ }^{16}16 in [90] where the action of a Schottky subgroup GGGGG
of SL2(R)SL2(R)SL_(2)(R)\mathrm{SL}_{2}(\mathbf{R})SL2(R) on the hyperbolic upper half-plane HHH\mathbb{H}H is considered. Lalley obtains that, for any x∈Hx∈Hx inHx \in \mathbb{H}x∈H, the number N(x,r)N(x,r)N(x,r)\mathcal{N}(x, r)N(x,r) of elements γγgamma\gammaγ of GGGGG such that
where dHdHd_(H)d_{\mathbb{H}}dH is hyperbolic distance, satisfies N(x,r)≈CeδrN(x,r)≈CeδrN(x,r)~~Ce^(delta r)\mathcal{N}(x, r) \approx C e^{\delta r}N(x,r)≈Ceδr, where δ=δ(G)δ=δ(G)delta=delta(G)\delta=\delta(G)δ=δ(G) is the Hausdorff dimension of the limit set of GGGGG, and C=C(G,x)>0C=C(G,x)>0C=C(G,x) > 0C=C(G, x)>0C=C(G,x)>0. Lalley's proof incorporates at various stages the following arguments:
Shell argument. By repeated application of a "renewal equation," the quantity N(x,r)N(x,r)N(x,r)\mathcal{N}(x, r)N(x,r) is related to a sum of N(y,r′)Ny,r′N(y,r^('))\mathcal{N}\left(y, r^{\prime}\right)N(y,r′), where the sum is over yyyyy on a shell of radius ≈cr≈cr~~cr\approx c r≈cr in a Cayley tree of GGGGG, and r′r′r^(')r^{\prime}r′ is a translate of rrrrr that corrects for the passage between xxxxx and yyyyy. The purpose of this shell argument is that now, the points yyyyy lie close to ∂H∂HdelH\partial \mathbb{H}∂H.
Passage to the boundary. Each of the resulting N(y,r′)Ny,r′N(y,r^('))\mathcal{N}\left(y, r^{\prime}\right)N(y,r′) is compared to an analogous quantity N∗(y∗,r′)N∗y∗,r′N^(**)(y^(**),r^('))\mathcal{N}^{*}\left(y^{*}, r^{\prime}\right)N∗(y∗,r′) where y∗y∗y^(**)y^{*}y∗ is a point in ∂H∂HdelH\partial \mathbb{H}∂H close to yyyyy. Because each yyyyy is close to ∂H∂HdelH\partial \mathbb{H}∂H, the errors incurred are acceptable.
Transfer operator techniques. Asymptotic formulas for N∗(y∗,r′)N∗y∗,r′N^(**)(y^(**),r^('))\mathcal{N}^{*}\left(y^{*}, r^{\prime}\right)N∗(y∗,r′) are obtained using the renewal method and spectral estimates for transfer operators. This gives asymptotic formulas for N(y,r′)Ny,r′N(y,r^('))\mathcal{N}\left(y, r^{\prime}\right)N(y,r′). The main terms of the asymptotic formulas satisfy recursive relationships between different yyyyy.
Recombination. One finally has to recombine all the asymptotic formulas obtained for N(y,r′)Ny,r′N(y,r^('))\mathcal{N}\left(y, r^{\prime}\right)N(y,r′) to obtain an asymptotic formula for N(x,r)N(x,r)N(x,r)\mathcal{N}(x, r)N(x,r). This is done using the recursive formulas obtained in the previous step.
Trying to follow the method outlined above for this orbital counting problem, we first need a suitable replacement for ∂H∂HdelH\partial \mathbb{H}∂H. Our idea is to use the projectivization of the hyperplane HHH\mathscr{H}H; we call this set ΔΔDelta\DeltaΔ. We compare points in the orbit of ΛΛLambda\LambdaΛ (generated by λjλjlambda_(j)\lambda_{j}λj in (5.4) to points in ΔΔDelta\DeltaΔ by taking logarithms of all coordinates and then projectivizing. This process does not necessarily lead to a point in ΔΔDelta\DeltaΔ; there is an important parameter α(z)=∏j=1n−2zjα(z)=âˆj=1n−2 zjalpha(z)=prod_(j=1)^(n-2)z_(j)\alpha(z)=\prod_{j=1}^{n-2} z_{j}α(z)=âˆj=1n−2zj that appears throughout the paper and measures how good the fit is. If α(z)α(z)alpha(z)\alpha(z)α(z) is large, then one can, in analogy with Lalley's setting, think of zzzzz as being "close to the boundary."
For Lalley, the word length of γγgamma\gammaγ is roughly proportional to the quantity dH(i,γx)−dH(i,γx)−d_(H)(i,gamma x)-d_{\mathbb{H}}(i, \gamma x)-dH(i,γx)−dH(i,x)dH(i,x)d_(H)(i,x)d_{\mathbb{H}}(i, x)dH(i,x) with respect to which he counts. This implies, during the shell argument, that all the elements of the shell are roughly the same distance from ∂H∂HdelH\partial \mathbb{H}∂H. However, for us, there are arbitrarily long words in the generators of ΛΛLambda\LambdaΛ for which α(z)α(z)alpha(z)\alpha(z)α(z) is small. We solve this problem using "acceleration," by replacing ΛΛLambda\LambdaΛ by Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′, and instead aim to follow Lalley's argument for orbits of Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′. This has the immediate benefit that we can guarantee that elements zzzzz of shells of radius LLLLL, with respect to Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′, have large α(z)α(z)alpha(z)\alpha(z)α(z), if we make LLLLL appropriately large.
However, the acceleration also has some costs to be paid. The first issue arising is that now Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′ has countably many generators, so shells for word length on Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′ are not finite. Instead of using shells, we use intersections of shells with the elements of the Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′-orbit whose coordinates are not too large. The second issue is that the original ΛΛLambda\LambdaΛ-orbit breaks up
FIGURE 16
In the same setting (n=4)(n=4)(n=4)(n=4)(n=4) of Figure 15 , we show in black the images of ΔΔDelta\DeltaΔ under the action of all words of length 10 in the generators {γ1,γ2,γ3}γ1,γ2,γ3{gamma_(1),gamma_(2),gamma_(3)}\left\{\gamma_{1}, \gamma_{2}, \gamma_{3}\right\}{γ1,γ2,γ3}.
into countably many Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′-orbits. So we not only have to perform the recombination argument for Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′, but then have to perform an extra summation over the countably many Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′-orbits.
After setting up our shell argument appropriately, we must perform the passage to the boundary (i.e., ΔΔDelta\DeltaΔ ). To this end, we compare orbits of Λ′Λ′Lambda^(')\Lambda^{\prime}Λ′ to orbits of Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′, where Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ is the linear semigroup. To get this to work, we must exploit the following "shadowing" feature of the map logloglog\loglog that takes logarithms of all entries of a vector. It says (roughly) that if log(z)logâ¡(z)log(z)\log (z)logâ¡(z) is within ϵϵepsilon\epsilonϵ of y∈Hy∈Hy inHy \in \mathscr{H}y∈H, with ϵϵepsilon\epsilonϵ on the scale of α(z)−2α(z)−2alpha(z)^(-2)\alpha(z)^{-2}α(z)−2, then for all λ∈Λ′,log(λ(z))λ∈Λ′,logâ¡(λ(z))lambda inLambda^('),log(lambda(z))\lambda \in \Lambda^{\prime}, \log (\lambda(z))λ∈Λ′,logâ¡(λ(z)) is within ϵϵepsilon\epsilonϵ of γ(log(z))γ(logâ¡(z))gamma(log(z))\gamma(\log (z))γ(logâ¡(z)), where γ∈Γ′γ∈Γ′gamma inGamma^(')\gamma \in \Gamma^{\prime}γ∈Γ′ is matched with λλlambda\lambdaλ in a natural way.
The completion of the proof relies on spectral estimates for transfer operators associated to the projective linear action of Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ on ΔΔDelta\DeltaΔ. There are three key issues arising here. First, to obtain the spectral estimates we need, we must establish that the action of Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ on ΔΔDelta\DeltaΔ is uniformly contracting; it is important to note that this argument would not work if the acceleration had not been performed previously. Secondly, we need to establish that the relevant "log-Jacobian" cocycle over the dynamical system is not cohomologous to a lattice cocycle. Finally, but importantly, we must obtain spectral estimates for transfer operators acting on C1(Δ)C1(Δ)C^(1)(Delta)C^{1}(\Delta)C1(Δ) which is accomplished by adapting Liverani's approach to spectral esti-
mates from [97]. See section 4 of [68], and references therein, for the discussion of Gauss map and Gauss measure [70,89[70,89[70,89[70,89[70,89 ] in this context.
The question of whether ββbeta\betaβ is irrational (Problem 2) remains a tantalizing open question, and one may wonder whether it is even algebraic. Our methods do give some partial insight into the nature of this mysterious number in terms of the action of Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′ on H/R+H/R+H//R_(+)\mathscr{H} / \mathbf{R}_{+}H/R+.
Theorem 10. The number ββbeta\betaβ is the unique parameter in (1,∞)(1,∞)(1,oo)(1, \infty)(1,∞) such that there exists a probability measure νβνβnu_(beta)\nu_{\beta}νβ on Δ=H/R+Δ=H/R+Delta=H//R_(+)\Delta=\mathscr{H} / \mathbf{R}_{+}Δ=H/R+with the property
∫w∈Δf(w)dvβ(w)=∑γ∈TΓ∫w∈Δf(γ⋅w)|Jacw(γ)|βn−1dvβ(w)∫w∈Δ f(w)dvβ(w)=∑γ∈TΓ ∫w∈Δ f(γ⋅w)Jacwâ¡(γ)βn−1dvβ(w)int_(w in Delta)f(w)dv_(beta)(w)=sum_(gamma inT_(Gamma))int_(w in Delta)f(gamma*w)|Jac_(w)(gamma)|^((beta)/(n-1))dv_(beta)(w)\int_{w \in \Delta} f(w) d v_{\beta}(w)=\sum_{\gamma \in T_{\Gamma}} \int_{w \in \Delta} f(\gamma \cdot w)\left|\operatorname{Jac}_{w}(\gamma)\right|^{\frac{\beta}{n-1}} d v_{\beta}(w)∫w∈Δf(w)dvβ(w)=∑γ∈TΓ∫w∈Δf(γ⋅w)|Jacwâ¡(γ)|βn−1dvβ(w)
for all f∈C0(Δ)f∈C0(Δ)f inC^(0)(Delta)f \in C^{0}(\Delta)f∈C0(Δ). We call νβνβnu_(beta)\nu_{\beta}νβ a conformal measure.
Theorem 10 can be viewed as a partial analog of the connection between the exponent of growth of a finitely generated Fuchsian group and the Hausdorff dimension of its limit set as a result of Patterson-Sullivan theory [117,138,139]. In our setting, the lack of any symmetric space means the parameter ββbeta\betaβ is not in any obvious way connected to the Hausdorff dimension of the compact Γ′Γ′Gamma^(')\Gamma^{\prime}Γ′-invariant subset of ΔΔDelta\DeltaΔ.
The issue of the existence of a single integral solution for general aaaaa and kkkkk is very subtle, even for n=3n=3n=3n=3n=3, as discussed in the next section.
6. HASSE PRINCIPLE ON SURFACES OF MARKOFF TYPE
Little is known about the values at integers assumed by affine cubic forms 17F17F^(17)F{ }^{17} F17F in three variables. For k≠0k≠0k!=0k \neq 0k≠0, set
The basic question is for which kkkkk is Vk,F(Z)≠∅Vk,F(Z)≠∅V_(k,F)(Z)!=O/V_{k, F}(\mathbb{Z}) \neq \emptysetVk,F(Z)≠∅, or, more generally, infinite or Zariski dense in Vk,FVk,FV_(k,F)V_{k, F}Vk,F ?
A prime example is F=SF=SF=SF=SF=S, the sum of three cubes,
There are obvious local congruence obstructions, namely that Vk,S(Z)=∅Vk,S(Z)=∅V_(k,S)(Z)=O/V_{k, S}(\mathbb{Z})=\emptysetVk,S(Z)=∅ if k≡4,5(mod9)k≡4,5(mod9)k-=4,5(mod9)k \equiv 4,5(\bmod 9)k≡4,5(mod9), but beyond that, it is possible that the answers to all three questions are yes for all the other kkkkk 's, which we call the admissible values (see [50,113]). It is known that strong approximation in its strongest form fails for Vk,S(Z)Vk,S(Z)V_(k,S)(Z)V_{k, S}(\mathbb{Z})Vk,S(Z); the global obstruction coming from an application of cubic reciprocity [41,49,75]). Moreover, the authors of [94] and [7] show that V1,S(Z)V1,S(Z)V_(1,S)(Z)V_{1, S}(\mathbb{Z})V1,S(Z) is Zariski dense in V1,SV1,SV_(1,S)V_{1, S}V1,S.
In [71] Ghosh and Sarnak investigate the Markoff form F=MF=MF=MF=MF=M,
17 By an affine form fffff in nnnnn variables we mean f∈Z[x1,…,xn]f∈Zx1,…,xnf inZ[x_(1),dots,x_(n)]f \in \mathbb{Z}\left[x_{1}, \ldots, x_{n}\right]f∈Z[x1,…,xn] whose leading homogeneous term f0f0f_(0)f_{0}f0 is nondegenerate and such that f−kf−kf-kf-kf−k is (absolutely) irreducible for all constants kkkkk.
FIGURE 17
Lattice points and fundamental set for k=3685k=3685k=3685k=3685k=3685.
FIGURE 18
Closeup of fundamental set for k=3685k=3685k=3685k=3685k=3685
Except for the case of the Cayley cubic with k=4,Vk;M(Z)k=4,Vk;M(Z)k=4,V_(k;M)(Z)k=4, V_{k ; M}(\mathbb{Z})k=4,Vk;M(Z) decomposes into a finite number hM(k)hM(k)h_(M)(k)\mathfrak{h}_{M}(k)hM(k) of ΓΓGamma\GammaΓ-orbits. For example, if k=0k=0k=0k=0k=0, then hM(k)=2hM(k)=2h_(M)(k)=2\mathfrak{h}_{M}(k)=2hM(k)=2 corresponds to the orbits of (0,0,0)(0,0,0)(0,0,0)(0,0,0)(0,0,0) and (3,3,3)(3,3,3)(3,3,3)(3,3,3)(3,3,3). In order to study hM(k)hM(k)h_(M)(k)\mathfrak{h}_{M}(k)hM(k) both theoretically and numerically, they give an explicit reduction (descent) for the action of ΓΓGamma\GammaΓ on Vk,M(Z)Vk,M(Z)V_(k,M)(Z)V_{k, M}(\mathbb{Z})Vk,M(Z). For this purpose, it is convenient to remove an explicit set of special admissible kkkkk 's, namely those for which there is a point in Vk,M(Z)Vk,M(Z)V_(k,M)(Z)V_{k, M}(\mathbb{Z})Vk,M(Z) with |xj|=0,1xj=0,1|x_(j)|=0,1\left|x_{j}\right|=0,1|xj|=0,1 or 2 . These kkkkk 's take the form (i) k=u2+v2k=u2+v2k=u^(2)+v^(2)k=u^{2}+v^{2}k=u2+v2, (ii) 4(k−1)=u2+3v24(k−1)=u2+3v24(k-1)=u^(2)+3v^(2)4(k-1)=u^{2}+3 v^{2}4(k−1)=u2+3v2, or (iii) k=4+u2k=4+u2k=4+u^(2)k=4+u^{2}k=4+u2. The number of these special kkkkk 's (referred to as exceptional) with 0≤k≤K0≤k≤K0 <= k <= K0 \leq k \leq K0≤k≤K is asymptotic to C′KlogKC′Klogâ¡KC^(')(K)/(sqrt(log K))C^{\prime} \frac{K}{\sqrt{\log K}}C′Klogâ¡K. The remaining admissible kkkkk 's are called generic (all negative admissible kkkkk 's are generic). For them Ghosh and Sarnak give the following elegant reduced forms:
FIGURE 19
Lattice points and fundamental set for k=−3691k=−3691k=-3691k=-3691k=−3691.
FIGURE 20
Closeup of fundamental set for k=−3691k=−3691k=-3691k=-3691k=−3691.
Proposition 11. (1) Let k≥5k≥5k >= 5k \geq 5k≥5 be generic and consider the compact set
This follows from the fact that when considering the values taken by the corresponding indefinite quadratic form in the yyyyy and zzzzz variables, for each fixed xxxxx, the units are bounded in number due to the restrictions imposed by the fundamental sets.
Let hM±(k)=|Fk±(Z)|hM±(k)=Fk±(Z)h_(M)^(+-)(k)=|F_(k)^(+-)(Z)|\mathfrak{h}_{M}^{ \pm}(k)=\left|\mathfrak{F}_{k}^{ \pm}(\mathbb{Z})\right|hM±(k)=|Fk±(Z)| where ±=sgn(k)±=sgnâ¡(k)+-=sgn(k)\pm=\operatorname{sgn}(k)±=sgnâ¡(k), this being defined for any kkkkk. Then for generic k,hM±(k)=hM(k)k,hM±(k)=hM(k)k,h_(M)^(+-)(k)=h_(M)(k)k, \mathfrak{h}_{M}^{ \pm}(k)=\mathfrak{h}_{M}(k)k,hM±(k)=hM(k) while otherwise hM(k)≤hM±(k)hM(k)≤hM±(k)h_(M)(k) <= h_(M)^(+-)(k)\mathfrak{h}_{M}(k) \leq \mathfrak{h}_{M}^{ \pm}(k)hM(k)≤hM±(k). We have
where C±>0C±>0C^(+-) > 0C^{ \pm}>0C±>0 and K→∞K→∞K rarr ooK \rightarrow \inftyK→∞.
So on average the numbers hM(k)hM(k)h_(M)(k)\mathfrak{h}_{M}(k)hM(k) are small. The explicit fundamental domains allow for the numerical computations; these indicate that
(6.5)∑0<k≤Kk admissible hM(k)=01∼C0Kθ(6.5)∑0<k≤Kk admissible hM(k)=0 1∼C0Kθ{:(6.5)sum_({:[0 < k <= K],[k" admissible "],[h_(M)(k)=0]:})1∼C_(0)K^(theta):}\begin{equation*}
\sum_{\substack{0<k \leq K \\ k \text { admissible } \\ \mathfrak{h}_{M}(k)=0}} 1 \sim C_{0} K^{\theta} \tag{6.5}
\end{equation*}(6.5)∑0<k≤Kk admissible hM(k)=01∼C0Kθ
with C0>0C0>0C_(0) > 0C_{0}>0C0>0 and θ≈0.8875…θ≈0.8875…theta~~0.8875 dots\theta \approx 0.8875 \ldotsθ≈0.8875…
The main result in [71] concerns the values assumed by MMMMM and the Hasse failures in (6.5):
Theorem 12. (i) There are infinitely many Hasse failures. More precisely, the number of 0<k≤K0<k≤K0 < k <= K0<k \leq K0<k≤K and −K≤k<0−K≤k<0-K <= k < 0-K \leq k<0−K≤k<0 for which the Hasse principle fails is at least K(logK)−14K(logâ¡K)−14sqrtK(log K)^(-(1)/(4))\sqrt{K}(\log K)^{-\frac{1}{4}}K(logâ¡K)−14 for KKKKK large.
(ii) Fix t≥0t≥0t >= 0t \geq 0t≥0. Then as K→∞K→∞K rarr ooK \rightarrow \inftyK→∞,
Part (ii) of the theorem is proved by comparing the number of points on Vk,M(Z)Vk,M(Z)V_(k,M)(Z)V_{k, M}(\mathbb{Z})Vk,M(Z) in certain tentacled regions gotten by special plane sections, with the expected number of solutions according to a product of local densities; the crucial point being that the variance of this comparison goes to zero on averaging |k|≤K|k|≤K|k| <= K|k| \leq K|k|≤K. As detailed in [71], this moving plane quadric method applies to more general cubic surfaces including those that do not carry morphisms.
ACKNOWLEDGMENTS
I would like to thank Jean Bourgain, Emmanuel Breuillard, Michael Magee, Igor Pak, Ryan Ronan, and Peter Sarnak. My various collaborations with them form the core of this report.
Figure 1 is courtesy of Matthew de Courcy-Ireland.
Figure 2 is courtesy of Elena Fuchs.
Figures 3-8 are courtesy of William Goldman.
Figures 9−149−149-149-149−14 are courtesy of Serge Cantat.
Figures 17-20 are courtesy of Amit Ghosh.
FUNDING
The author was supported, in part, by NSF award DMS-1603715.
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ALEXANDER GAMBURD
The Graduate Center, CUNY, New York, NY, USA, agamburd @ gmail.com
THE NUMBER OF RATIONAL POINTS ON A CURVE OF GENUS AT LEAST TWO
PHILIPP HABEGGER
ABSTRACT
The Mordell Conjecture states that a smooth projective curve of genus at least 2 defined over number field FFFFF admits only finitely many FFFFF-rational points. It was proved by Faltings in the 1980s and again using a different strategy by Vojta. Despite there being two different proofs of the Mordell Conjecture, many important questions regarding the set of FFFFF-rational points remain open. This survey concerns recent developments towards upper bounds on the number of rational points in connection with a question of Mazur.
for certain bivariate polynomials P∈Q[X,Y]P∈Q[X,Y]P inQ[X,Y]P \in \mathbb{Q}[X, Y]P∈Q[X,Y].
To make the statement and results precise, we will adopt the language of projective algebraic curves. Indeed, for the study of the zero set, we may assume that PPPPP is irreducible, even as a polynomial in C[X,Y]C[X,Y]C[X,Y]\mathbb{C}[X, Y]C[X,Y]. Moreover, its homogenization defines a projective curve in the projective plane. The classical procedure of normalization allows us to resolve any singularities. The result is an irreducible smooth projective curve defined over QQQ\mathbb{Q}Q. Its complex points define a compact Riemann surface of genus g∈{0,1,2,…}g∈{0,1,2,…}g in{0,1,2,dots}g \in\{0,1,2, \ldots\}g∈{0,1,2,…}.
Conversely, let us assume we are presented with a smooth projective curve CCCCC defined over QQQ\mathbb{Q}Q that is irreducible as a curve taken over CCC\mathbb{C}C. The genus ggggg of C(C)C(C)C(C)C(\mathbb{C})C(C) taken as a Riemann surface has important consequences for arithmetic questions on C(Q)C(Q)C(Q)C(\mathbb{Q})C(Q). Indeed, Mordell's Conjecture, proved by Faltings [25], states that #C(Q)#C(Q)#C(Q)\# C(\mathbb{Q})#C(Q) is finite if g≥2g≥2g >= 2g \geq 2g≥2.
We begin by formulating the Mordell Conjecture in slightly higher generality. We then discuss the history of results towards this conjecture. Finally, we give an overview of the proof of a joint work by Ziyang Gao, Vesselin Dimitrov, and the author towards a question of Mazur regarding upper bounds for the cardinality #C (Q)(Q)(Q)(\mathbb{Q})(Q). The upper bound will depend on the genus ggggg and the Mordell-Weil rank of the Jacobian of CCCCC. For a special case of this result that does not make reference to Jacobians, we refer to Section 6.
1.1. The Mordell Conjecture
We begin by recalling Faltings's Theorem [25], a finiteness statement originally conjectured by Mordell [48]. By a curve we mean a geometrically irreducible projective variety of dimension 1. Throughout, we let FFFFF denote a number field and F¯F¯bar(F)\bar{F}F¯ a fixed algebraic closure of FFFFF.
Theorem 1.1 (Faltings [25]). Let CCCCC be a smooth curve of genus at least 2 defined over a number field FFFFF. Then C(F)C(F)C(F)C(F)C(F) is finite.
If the genus of CCCCC is small, then one cannot expect finiteness. Indeed, the set C(F)C(F)C(F)C(F)C(F) is nonempty after replacing FFFFF by a suitable finite extension. If CCCCC has genus 0 , then CCCCC is isomorphic to the projective line and thus C(F)C(F)C(F)C(F)C(F) is infinite. If CCCCC has genus 1, then CCCCC together with a point in C(F)C(F)C(F)C(F)C(F) is an elliptic curve. In particular, we obtain an algebraic group. After possibly extending FFFFF again, we may assume that C(F)C(F)C(F)C(F)C(F) contains a point of infinite order. So C(F)C(F)C(F)C(F)C(F) is infinite.
To prove the Mordell Conjecture, Faltings first proved the Shafarevich Conjecture for abelian varieties. At the time, the latter was known to imply the Mordell Conjecture thanks to a construction of Kodaira-Parshin.
Later, Vojta [62] gave a different proof of the Mordell Conjecture that is rooted in diophantine approximation. Bombieri [8] then simplified Vojta's proof. We will recall Vojta's
approach for curves in Section 3. The technical heart is the Vojta inequality which we formulate below as Theorem 3.1.
Faltings generalized Vojta's proof of the Mordell Conjecture to cover subvarieties of any dimension of an abelian variety. Indeed, Faltings [26, 27] and Hindry [36] proved the Mordell-Lang Conjecture for subvarieties of abelian varieties. Let AAAAA be an abelian variety defined over FFFFF and suppose ΓΓGamma\GammaΓ is a subgroup of A(F¯)A(F¯)A( bar(F))A(\bar{F})A(F¯). The division closure of ΓΓGamma\GammaΓ is the subgroup
{P∈A(F¯) : there exists an integer n≥1 with nP∈Γ}{P∈A(F¯) : there exists an integer n≥1 with nP∈Γ}{P in A( bar(F))" : there exists an integer "n >= 1" with "nP in Gamma}\{P \in A(\bar{F}) \text { : there exists an integer } n \geq 1 \text { with } n P \in \Gamma\}{P∈A(F¯) : there exists an integer n≥1 with nP∈Γ}
of A(F¯)A(F¯)A( bar(F))A(\bar{F})A(F¯). For example, the division closure of the trivial subgroup Γ={0}Γ={0}Gamma={0}\Gamma=\{0\}Γ={0} is the subgroup Ators Ators A_("tors ")A_{\text {tors }}Ators of all points of finite order of A(F¯)A(F¯)A( bar(F))A(\bar{F})A(F¯). The following theorem holds for all base fields of characteristic 0 .
The special case when Γ=Ators Γ=Ators Gamma=A_("tors ")\Gamma=A_{\text {tors }}Γ=Ators is called the Manin-Mumford Conjecture and was proved by Raynaud [53].
More recently, Lawrence and Venkatesh [41] gave yet another proof of the Mordell Conjecture. It was inspired by Faltings's original approach and the method of ChabautyKim. We refer to the survey [6] on these developments.
In this survey we concentrate mainly on the case of curves and comment on possible extensions to the higher dimensional case.
1.2. Some remarks on effectivity
Despite the variety of approaches to the Mordell Conjecture, no effective proof is known. For example, if the curve CCCCC is presented explicitly as the vanishing locus of homogeneous polynomial equations with rational coefficients, say, then in full generality we know no algorithm that produces the finite list of rational points of CCCCC. The question of effectivity is already open in genus 2 , for example, for the family Y2=X5+tY2=X5+tY^(2)=X^(5)+tY^{2}=X^{5}+tY2=X5+t parametrized by ttttt. Proving an effective version of the Mordell Conjecture is among the most important outstanding problems in diophantine geometry.
Although no general algorithm that determines the set of rational points is currently known, it is sometimes possible to determine the set of rational points. For example, we refer to the Chabauty-Coleman method [13,15][13,15][13,15][13,15][13,15] which provides a clean upper bound for the number of rational points subject to a hypothesis on the Mordell-Weil rank of the Jacobian of CCCCC. In several applications, this bound equals a lower bound for the number of rational points coming from a list of known rational points. Moreover, aspects of Kim's generalization of the Chabauty method were used by Balakrishnan, Dogra, Müller, Tuitman, and Vonk [5] to compute all rational points of the split Cartan modular curve of level 13 which appears in relation to Serre's uniformity question. A different approach motivated by work of Dem-
janenko and the theory of unlikely intersections was developed in a program by Checcoli, Veneziano, Viada [14]. Here too a condition on the rank of the curve's Jacobian is required for the method to apply. An remarkable aspect to this approach is that the authors obtain an explicit upper bound for the height of a rational point.
1.3. The number of rational points: conjectures and results
Given CCCCC and FFFFF as in Theorem 1.1, which invariants of CCCCC need to appear in an upper bound for #C(F)?
Example 1.3. (i) Consider the hyperelliptic curve CCCCC presented by
Its genus equals (2022−2)/2=1010(2022−2)/2=1010(2022-2)//2=1010(2022-2) / 2=1010(2022−2)/2=1010. Then CCCCC contains the rational points (1,0),…,(2022,0)(1,0),…,(2022,0)(1,0),dots,(2022,0)(1,0), \ldots,(2022,0)(1,0),…,(2022,0). Together with the two points at infinity, we obtain at least 2024 rational points. This example easily generalizes to higher genus. For any g≥2g≥2g >= 2g \geq 2g≥2 and square-free f∈Q[X]f∈Q[X]f inQ[X]f \in \mathbb{Q}[X]f∈Q[X] of degree 2g+22g+22g+22 g+22g+2, the equation y2=f(x)y2=f(x)y^(2)=f(x)y^{2}=f(x)y2=f(x) determines a hyperelliptic curve CCCCC of genus ggggg. If fffff splits into (pairwise distinct) linear factors over QQQ\mathbb{Q}Q, then #C(Q)≥2g+4#C(Q)≥2g+4#C(Q) >= 2g+4\# C(\mathbb{Q}) \geq 2 g+4#C(Q)≥2g+4. So any upper bound for #C(Q)#C(Q)#C(Q)\# C(\mathbb{Q})#C(Q) must depend on the genus.
This lower bound is far from the truth. Stoll discovered a genus 2 curve defined over QQQ\mathbb{Q}Q with 642 rational points in a family of such curves constructed by Elkies. Mestre showed that for all g≥2g≥2g >= 2g \geq 2g≥2 there is a smooth curve of genus ggggg defined over QQQ\mathbb{Q}Q with at least 8g+168g+168g+168 g+168g+16 rational points.
(ii) Let us now fix the curve CCCCC and let the number field FFFFF vary. We take CCCCC as the genus 2 hyperelliptic curve presented by y2=x5+1y2=x5+1y^(2)=x^(5)+1y^{2}=x^{5}+1y2=x5+1. Consider an integer n≥0n≥0n >= 0n \geq 0n≥0 and the points {(m,±(m5+1)1/2):m∈{0,…,n}}m,±m5+11/2:m∈{0,…,n}{(m,+-(m^(5)+1)^(1//2)):m in{0,dots,n}}\left\{\left(m, \pm\left(m^{5}+1\right)^{1 / 2}\right): m \in\{0, \ldots, n\}\right\}{(m,±(m5+1)1/2):m∈{0,…,n}}. So C(F)C(F)C(F)C(F)C(F) has at least 2n+2+12n+2+12n+2+12 n+2+12n+2+1 elements where F=Q((m5+1)1/2)m∈{1,…,n}F=Qm5+11/2m∈{1,…,n}F=Q((m^(5)+1)^(1//2))_(m in{1,dots,n})F=\mathbb{Q}\left(\left(m^{5}+1\right)^{1 / 2}\right)_{m \in\{1, \ldots, n\}}F=Q((m5+1)1/2)m∈{1,…,n}. Any upper bound C(F)C(F)C(F)C(F)C(F), even for CCCCC fixed, must depend on FFFFF.
Gabriel Dill pointed out that the number of FFFFF-points grows at least logarithmically in the degree [F:Q][F:Q][F:Q][F: \mathbb{Q}][F:Q] in this case. Indeed, [F:Q]≤2n[F:Q]≤2n[F:Q] <= 2^(n)[F: \mathbb{Q}] \leq 2^{n}[F:Q]≤2n, so #C F(F)≥2n≥F(F)≥2n≥F(F) >= 2n >=F(F) \geq 2 n \geqF(F)≥2n≥2(log[F:Q])/log22(logâ¡[F:Q])/logâ¡22(log[F:Q])//log 22(\log [F: \mathbb{Q}]) / \log 22(logâ¡[F:Q])/logâ¡2.
Let us consider the modular curve X0(37)X0(37)X_(0)(37)X_{0}(37)X0(37) which has genus 2 and is defined over QQQ\mathbb{Q}Q. Let ppppp be one of the infinitely many prime numbers for which the Legendre symbol satisfies (−p/37)=1(−p/37)=1(-p//37)=1(-p / 37)=1(−p/37)=1; so 37 splits in the quadratic field K=Q(−p)K=Q(−p)K=Q(sqrt(-p))K=\mathbb{Q}(\sqrt{-p})K=Q(−p). Let FFFFF denote the Hilbert Class Field of KKKKK. There is an elliptic curve EEEEE defined over FFFFF with complex multiplication by the ring of integers of KKKKK. Moreover, EEEEE admits an isogeny of degree 37 to an elliptic curve defined over FFFFF. Thus X0X0X_(0)X_{0}X0 (37) has an FFFFF-rational point. The Galois group Gal(F/K)Galâ¡(F/K)Gal(F//K)\operatorname{Gal}(F / K)Galâ¡(F/K) acts on the FFFFF-rational points of X0(37)X0(37)X_(0)(37)X_{0}(37)X0(37). It is also known to act transitively on the moduli of elliptic curves with the same endomorphism ring as EEEEE. Thus #X0(37)(F)#X0(37)(F)#X_(0)(37)(F)\# X_{0}(37)(F)#X0(37)(F) is no less than [F:K][F:K][F:K][F: K][F:K] which equals the class number of KKKKK by Class Field Theory. So #X0(37)(F)≥[F:K]=[F:Q]/2#X0(37)(F)≥[F:K]=[F:Q]/2#X_(0)(37)(F) >= [F:K]=[F:Q]//2\# X_{0}(37)(F) \geq[F: K]=[F: \mathbb{Q}] / 2#X0(37)(F)≥[F:K]=[F:Q]/2. By the Landau-
Siegel Theorem, [F:Q]→∞[F:Q]→∞[F:Q]rarr oo[F: \mathbb{Q}] \rightarrow \infty[F:Q]→∞ as p→∞p→∞p rarr oop \rightarrow \inftyp→∞. In particular, any upper bound for #X0(37)(F)#X0(37)(F)#X_(0)(37)(F)\# X_{0}(37)(F)#X0(37)(F) must grow at least linearly in [F:Q][F:Q][F:Q][F: \mathbb{Q}][F:Q].
The Uniformity Conjecture by Caporaso-Harris-Mazur [11] predicts that the genus and base field of a curve are the only invariants required for a general upper bound.
Conjecture 1.4 (Caporaso-Harris-Mazur). Let g≥2g≥2g >= 2g \geq 2g≥2 be an integer and FFFFF a number field. There exists c(g,F)≥1c(g,F)≥1c(g,F) >= 1c(g, F) \geq 1c(g,F)≥1 such that if CCCCC is a smooth curve of genus ggggg defined over FFFFF, then #C(F)≤c(g,F)#C(F)≤c(g,F)#C(F) <= c(g,F)\# C(F) \leq c(g, F)#C(F)≤c(g,F).
Mazur [46] posed the following question, which is a weaker version of the Uniformity Conjecture. We let Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C) denote the Jacobian of a smooth curve CCCCC defined over a field. Then Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C) is a principally polarized abelian variety whose dimension equals the genus of CCCCC. If the base field is a number field FFFFF, then Jac(C)(F)Jacâ¡(C)(F)Jac(C)(F)\operatorname{Jac}(C)(F)Jacâ¡(C)(F) is a finitely generated abelian group by the Mordell-Weil Theorem.
Question 1.5 (Mazur [46, P. 223]). Let g≥2g≥2g >= 2g \geq 2g≥2 and rrrrr be integers and let FFFFF be a number field. There exists c(g,r,F)≥1c(g,r,F)≥1c(g,r,F) >= 1c(g, r, F) \geq 1c(g,r,F)≥1 such that if CCCCC is a smooth curve of genus ggggg defined over FFFFF such that the rank of the Mordell-Weil group satisfies rk Jac(C)(F)≤rJacâ¡(C)(F)≤rJac(C)(F) <= r\operatorname{Jac}(C)(F) \leq rJacâ¡(C)(F)≤r, then #C(F)≤c(g,r,F)#C(F)≤c(g,r,F)#C(F) <= c(g,r,F)\# C(F) \leq c(g, r, F)#C(F)≤c(g,r,F).
Let us review some work on upper bounds for #C(F)#C(F)#C(F)\# C(F)#C(F). Parshin [59] showed how to extract an upper bound for the number of rational points from Faltings's theorem. In his original paper, Vojta [62] gave a blueprint on how to bound from above the number of rational points for a general CCCCC. This bound was refined by Bombieri [8] and de Diego [19]. However, these works did not provide an answer to Mazur's question.
js:Cs→Jac(Cs) given by P↦((2g−2)[P]−Ks)/∼js:Cs→Jacâ¡Cs given by P↦(2g−2)[P]−Ks/∼j_(s):C_(s)rarr Jac(C_(s))quad" given by "P|->((2g-2)[P]-K_(s))//∼j_{s}: \mathscr{C}_{s} \rightarrow \operatorname{Jac}\left(\mathscr{C}_{s}\right) \quad \text { given by } P \mapsto\left((2 g-2)[P]-\mathcal{K}_{s}\right) / \simjs:Cs→Jacâ¡(Cs) given by P↦((2g−2)[P]−Ks)/∼
Let θsθstheta_(s)\theta_{s}θs denote the theta divisor on Jac(Cs)Jacâ¡CsJac(C_(s))\operatorname{Jac}\left(\mathscr{C}_{s}\right)Jacâ¡(Cs) and h^s=h^Cs,θsh^s=h^Cs,θshat(h)_(s)= hat(h)_(C_(s),theta_(s))\hat{h}_{s}=\hat{h}_{\mathscr{C}_{s}, \theta_{s}}h^s=h^Cs,θs the canonical height on Jac(Cs)Jacâ¡CsJac(C_(s))\operatorname{Jac}\left(\mathscr{C}_{s}\right)Jacâ¡(Cs) attached to this divisor.
Theorem 1.6 (de Diego [19]). There exists c(C)>1c(C)>1c(C) > 1c(\mathscr{C})>1c(C)>1 such that if F′/FF′/FF^(')//FF^{\prime} / FF′/F is a finite extension and s∈S(F′)s∈SF′s in S(F^('))s \in S\left(F^{\prime}\right)s∈S(F′), then
Roughly speaking, this theorem tells us that the number of points of CsCsC_(s)\mathscr{C}_{s}Cs of sufficiently large canonical height is bounded as in Mazur's question. We will often call these points large points. It is striking that the constant 7 is admissible for all genera; a fact that already appeared in Bombieri's work [8]. For smooth curves of genus 2 defined over QQQ\mathbb{Q}Q with a marked Weierstrass point, Alpoge [2] improved 7 to 1.872 .
is finite by the Northcott property for height functions which we will review in Section 2. To obtain a positive answer to Mazur's question we need, roughly speaking, to get a similar bound as in Theorem 1.6 for the cardinality of (1.1). There are quantitative versions of Northcott's Theorem. Estimating the cardinality (1.1) with these does, however, introduce a dependence on h(s)h(s)h(s)h(s)h(s).
Here degCdegâ¡Cdeg C\operatorname{deg} Cdegâ¡C is the degree of CCCCC with respect to the principal polarization. Moreover, h0(A)h0(A)h_(0)(A)h_{0}(A)h0(A) is a height of the abelian variety AAAAA whose definition involves classical theta functions and the degree [F:Q][F:Q][F:Q][F: \mathbb{Q}][F:Q]. Roughly speaking, h0(A)h0(A)h_(0)(A)h_{0}(A)h0(A) encodes a bound for the coefficients needed to reconstruct the abelian variety AAAAA. Mazur's question does not allow for a dependence on h0(A)h0(A)h_(0)(A)h_{0}(A)h0(A). The hypothesis that CCCCC is not smooth of genus 1 is natural and cannot be dropped in general. It is equivalent to stating that CCCCC is not a translate of an algebraic subgroup of AAAAA.
David and Philippon's approach to Mazur's question and its higher-dimensional counterparts is via a strong quantitative version of the Bogomolov Conjecture on points of small height. A suitable version is Conjecture 1.5 [18] where the lower bound for the canonical height grows linearly in the Faltings height. We refer to [18, THÉORĖME 1.11] regarding the connection to rational points and more generally the Mordell-Lang Conjecture.
David and Philippon were able to strengthen their height lower bound when AAAAA is a power of an elliptic curve. This provided more evidence towards a positive answer for Mazur's question. Here is a version of their result for curves; their general result holds for subvarieties of a power of an elliptic curve.
Thanks to a specialization argument, David and Philippon extended the above result to include the case where FFFFF is an arbitrary field of characteristic 0 . David, Nakamaye, and Philippon [16] then proved the existence of a (g−2)(g−2)(g-2)(g-2)(g−2)-dimensional family of curves of genus ggggg for which Mazur's question has a positive answer.
We now very briefly turn to some cardinality estimates using the Chabauty-Coleman method, which is based on ppppp-adic analysis. It can produce finiteness of C(F)C(F)C(F)C(F)C(F) with a clean cardinality estimate subject to a restriction on the rank of the Mordell-Weil group.
Theorem 1.9 (Coleman [15]). Suppose CCCCC is a smooth curve of genus g≥2g≥2g >= 2g \geq 2g≥2 defined over QQQ\mathbb{Q}Q with rkac(C)(Q)≤g−1rkacâ¡(C)(Q)≤g−1rkac(C)(Q) <= g-1\operatorname{rkac}(C)(\mathbb{Q}) \leq g-1rkacâ¡(C)(Q)≤g−1. If p>2gp>2gp > 2gp>2 gp>2g is a prime number where CCCCC has good reduction C~C~tilde(C)\tilde{C}C~, then #C(Q)≤2g−2+#C~(Fp)#C(Q)≤2g−2+#C~Fp#C(Q) <= 2g-2+# tilde(C)(F_(p))\# C(\mathbb{Q}) \leq 2 g-2+\# \tilde{C}\left(\mathbb{F}_{p}\right)#C(Q)≤2g−2+#C~(Fp).
In combination with the Hasse-Weil bound #C~(Fp)≤p+1+2gp#C~Fp≤p+1+2gp# tilde(C)(F_(p)) <= p+1+2gsqrtp\# \tilde{C}\left(\mathbb{F}_{p}\right) \leq p+1+2 g \sqrt{p}#C~(Fp)≤p+1+2gp, the estimate above yields a bound for #C(Q)#C(Q)#C(Q)\# C(\mathbb{Q})#C(Q) in terms of ggggg and ppppp alone. Observe that a dependence in the arithmetic of CCCCC appears through the prime ppppp. Stoll was able to remove this dependence for hyperelliptic curves at the cost of a stronger restriction on the rank of the Mordell-Weil group.
Theorem 1.10 (Stoll [58]). Let g≥2g≥2g >= 2g \geq 2g≥2 and d≥1d≥1d >= 1d \geq 1d≥1 be integers. There exists c(g,d)>0c(g,d)>0c(g,d) > 0c(g, d)>0c(g,d)>0 with the following property. Suppose CCCCC is a smooth hyperelliptic curve of genus ggggg defined over FFFFF with [F:Q]≤d[F:Q]≤d[F:Q] <= d[F: \mathbb{Q}] \leq d[F:Q]≤d. If rkJac(C)(Q)≤g−3rkâ¡Jacâ¡(C)(Q)≤g−3rk Jac(C)(Q) <= g-3\operatorname{rk} \operatorname{Jac}(C)(\mathbb{Q}) \leq g-3rkâ¡Jacâ¡(C)(Q)≤g−3, then #C(F)≤c(g,d)#C(F)≤c(g,d)#C(F) <= c(g,d)\# C(F) \leq c(g, d)#C(F)≤c(g,d).
Later, Katz, Rabinoff, and Zureick-Brown dropped the hyperellipticity condition.
Theorem 1.11 (Katz, Rabinoff, and Zureick-Brown [38]). Let g≥2g≥2g >= 2g \geq 2g≥2 and d≥1d≥1d >= 1d \geq 1d≥1 be integers. There exists c(g,d)>0c(g,d)>0c(g,d) > 0c(g, d)>0c(g,d)>0 with the following property. Suppose CCCCC is a smooth curve of genus ggggg defined over FFFFF with [F:Q]≤d[F:Q]≤d[F:Q] <= d[F: \mathbb{Q}] \leq d[F:Q]≤d. If rkJac(C)(Q)≤g−3rkâ¡Jacâ¡(C)(Q)≤g−3rk Jac(C)(Q) <= g-3\operatorname{rk} \operatorname{Jac}(C)(\mathbb{Q}) \leq g-3rkâ¡Jacâ¡(C)(Q)≤g−3, then #C(F)≤c(g,d)#C(F)≤c(g,d)#C(F) <= c(g,d)\# C(F) \leq c(g, d)#C(F)≤c(g,d).
After this detour to the Chabauty-Coleman method, we return to Vojta's method. Vesselin Dimitrov, Ziyang Gao, and the author have recently proved a lower bound for the canonical height that can be used as a replacement for the lower bounds by David and Philippon [17,18][17,18][17,18][17,18][17,18] in the context of Mordell's Conjecture. We recall this height inequality in Section 4.2 below. Indeed, it led to a positive answer to a strengthening of Mazur's question. The following result is new already in genus 2 .
Theorem 1.12 (Dimitrov, Gao, and Habegger [24, theorem 1.1]). Let g≥2g≥2g >= 2g \geq 2g≥2 and d≥1d≥1d >= 1d \geq 1d≥1 be integers, there exist c′(g,d)>1c′(g,d)>1c^(')(g,d) > 1c^{\prime}(g, d)>1c′(g,d)>1 and c(g,d)>1c(g,d)>1c(g,d) > 1c(g, d)>1c(g,d)>1 with the following property. Suppose CCCCC is a smooth curve of genus ggggg defined over a number field FFFFF such that [F:Q]≤d[F:Q]≤d[F:Q] <= d[F: \mathbb{Q}] \leq d[F:Q]≤d. Then
Regarding the Caporaso-Harris-Mazur Uniformity Conjecture, we ask
Question 1.13. Can the cardinality #C(F)#C(F)#C(F)\# C(F)#C(F) be bounded from above by a function that is polynomial in [F:Q][F:Q][F:Q][F: \mathbb{Q}][F:Q] and ggggg ?
No one currently knows an algorithm that computes the rank of the Mordell-Weil group Jac(C)(F)Jacâ¡(C)(F)Jac(C)(F)\operatorname{Jac}(C)(F)Jacâ¡(C)(F). However, upper bounds for this rank follow, for example, from the OoeTop Theorem [49]. We discuss this in more depth in Section 6.
Our results also cover points on CCCCC that lie in the division closure of a finitely generated subgroup. Let Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ denote the algebraic closure of QQQ\mathbb{Q}Q in CCC\mathbb{C}C. The Jacobian Jac (C)(C)(C)(C)(C) of a smooth curve CCCCC of genus ggggg defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ corresponds to a Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯-point of the coarse moduli space AgAgA_(g)\mathbb{A}_{g}Ag of ggggg-dimensional principally polarized abelian varieties. We let [Jac(C)][Jacâ¡(C)][Jac(C)][\operatorname{Jac}(C)][Jacâ¡(C)] denote the point of Ag(Q¯)Ag(Q¯)A_(g)( bar(Q))\mathbb{A}_{g}(\overline{\mathbb{Q}})Ag(Q¯) corresponding to Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C) with its canonical principal polarization.
For example, if g=1g=1g=1g=1g=1 then Ag=A1Ag=A1A_(g)=A^(1)\mathbb{A}_{g}=\mathbb{A}^{1}Ag=A1 is the affine line. If EEEEE is an elliptic curve defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯, then [E][E][E][E][E] is the jjjjj-invariant of EEEEE.
In general, AgAgA_(g)\mathbb{A}_{g}Ag is a quasiprojective variety of dimension g(g+1)/2g(g+1)/2g(g+1)//2g(g+1) / 2g(g+1)/2 defined over QQQ\mathbb{Q}Q. We may fix an immersion ι:Ag↪Pnι:Ag↪Pniota:A_(g)↪P^(n)\iota: \mathbb{A}_{g} \hookrightarrow \mathbb{P}^{n}ι:Ag↪Pn into projective space. Then the absolute logarithmic Weil height hhhhh, see Section 2 for a definition, pulls back to a function h∘ι:Ag(Q¯)→[0,∞)h∘ι:Ag(Q¯)→[0,∞)h@iota:A_(g)( bar(Q))rarr[0,oo)h \circ \iota: \mathbb{A}_{g}(\overline{\mathbb{Q}}) \rightarrow[0, \infty)h∘ι:Ag(Q¯)→[0,∞).
If CCCCC is a smooth curve of genus g≥1g≥1g >= 1g \geq 1g≥1 defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ and if P0∈C(Q¯)P0∈C(Q¯)P_(0)in C( bar(Q))P_{0} \in C(\overline{\mathbb{Q}})P0∈C(Q¯), then a point P∈C(Q¯)P∈C(Q¯)P in C( bar(Q))P \in C(\overline{\mathbb{Q}})P∈C(Q¯) defines a divisor [P]−[P0][P]−P0[P]-[P_(0)][P]-\left[P_{0}\right][P]−[P0] of degree 0 . One obtains a closed immersion
C↪Jac(C) from P↦([P]−[P0])/∼C↪Jacâ¡(C) from P↦[P]−P0/∼C↪Jac(C)quad" from "P|->([P]-[P_(0)])//∼C \hookrightarrow \operatorname{Jac}(C) \quad \text { from } P \mapsto\left([P]-\left[P_{0}\right]\right) / \simC↪Jacâ¡(C) from P↦([P]−[P0])/∼
where ∼∼∼\sim∼ again denotes linear equivalence, induces a closed immersion. We will write C−P0C−P0C-P_(0)C-P_{0}C−P0 for the image of CCCCC in Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C).
Theorem 1.14 (Dimitrov, Gao, and Habegger [24, THEOREM 1.2]). Let g≥2g≥2g >= 2g \geq 2g≥2 be an integer. There exist c(g,ι)>1,c′(g,ι)>0c(g,ι)>1,c′(g,ι)>0c(g,iota) > 1,c^(')(g,iota) > 0c(g, \iota)>1, c^{\prime}(g, \iota)>0c(g,ι)>1,c′(g,ι)>0, and c′′(g,ι)>0c′′(g,ι)>0c^('')(g,iota) > 0c^{\prime \prime}(g, \iota)>0c′′(g,ι)>0 that depend on ggggg and the immersion ιιiota\iotaι with the following property. Suppose CCCCC is a smooth curve of genus ggggg defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ and let P0∈C(Q¯)P0∈C(Q¯)P_(0)in C( bar(Q))P_{0} \in C(\overline{\mathbb{Q}})P0∈C(Q¯). Let ΓΓGamma\GammaΓ be the division closure of a finitely generated subgroup of Jac(C)(Q¯)Jacâ¡(C)(Q¯)Jac(C)( bar(Q))\operatorname{Jac}(C)(\overline{\mathbb{Q}})Jacâ¡(C)(Q¯) of rank rrrrr. If
In particular, we may take Γ=Jac(C)tors Γ=Jacâ¡(C)tors Gamma=Jac(C)_("tors ")\Gamma=\operatorname{Jac}(C)_{\text {tors }}Γ=Jacâ¡(C)tors and r=0r=0r=0r=0r=0. Thus the theorem yields a uniform bound for the number of torsion points that lie on C−P0C−P0C-P_(0)C-P_{0}C−P0 if the height of ι([Jac(C)])ι([Jacâ¡(C)])iota([Jac(C)])\iota([\operatorname{Jac}(C)])ι([Jacâ¡(C)]) is sufficiently large.
Suppose that CCCCC is defined over a number field FFFFF. Then [Jac(C)][Jacâ¡(C)][Jac(C)][\operatorname{Jac}(C)][Jacâ¡(C)] is an FFFFF-rational point of the moduli space AgAgA_(g)\mathbb{A}_{g}Ag. If we impose also h(ι([Jac(C)]))<c′′(g,ι)h(ι([Jacâ¡(C)]))<c′′(g,ι)h(iota([Jac(C)])) < c^('')(g,iota)h(\iota([\operatorname{Jac}(C)]))<c^{\prime \prime}(g, \iota)h(ι([Jacâ¡(C)]))<c′′(g,ι), then [Jac(C)][Jacâ¡(C)][Jac(C)][\operatorname{Jac}(C)][Jacâ¡(C)] lies in a finite set by the Northcott property. Thus Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C) is in one of at most finitely many Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯-isomorphism classes and so is CCCCC by the Torelli Theorem.
Raynaud proved the following result which is the Manin-Mumford Conjecture for curves.
Using a different approach involving equidistribution and motivated by dynamical systems, DeMarco, Krieger, and Ye [20] had made substantial progress towards the Uniform Manin-Mumford Conjecture. They proved it for smooth curves of genus 2 defined over CCC\mathbb{C}C that are double covers of an elliptic curve when the base point P0P0P_(0)P_{0}P0 is a Weierstrass point.
In a preprint, Kühne [39] complemented the method in [24] using ideas from equidistribution to prove the Uniform Manin-Mumford Conjecture.
In contrast to Theorem 1.14, Kühne is able to handle curves CCCCC defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ for which [Jac(C)][Jacâ¡(C)][Jac(C)][\operatorname{Jac}(C)][Jacâ¡(C)] has height less than c′′(g,ι)c′′(g,ι)c^('')(g,iota)c^{\prime \prime}(g, \iota)c′′(g,ι). Once uniformity is established for all curves over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯, Kühne is able to pass to the base field CCC\mathbb{C}C using a specialization argument laid out by Dimitrov, Gao, and the author [22] which relies on a result of Masser [43]. Kühne thus answers an older question of Mazur, see the top of page 234 [45], and obtains the full Mordell-Lang variant for curves.
DeMarco and Mavraki's [21] work on a relative version of the Bogomolov conjecture, see [72] and [22], motivated Kühne [39,40] to extend the reach of Arakelovian equidistribution methods of Szpiro-Ullmo-Zhang [60] and Yuan [65] to families of abelian varieties over a quasiprojective base. For algebraic curves, this settles the uniform Bogomolov and the uniform Manin-Mumford conjectures.
Yuan [66] recently gave another proof of Theorem 1.16. His method also runs via a uniform Bogomolov theorem and thus contains aspects related to height lower bounds. However, Yuan's approach relies on arithmetic bigness, rather than on equidistribution. It is independent of the approaches mentioned above and uses a new theory of adelic line bundles over quasiprojective varieties developed by Yuan and Zhang [67] which generalizes Zhang's theory [70] in the projective case. They derive a height inequality for a polarized dynamical system, see Theorem 1.3.2 and Section 6 [67], that extends our own bound. One aspect of Yuan's method is that it works for global fields in any characteristic.
We come to some questions regarding the base constant c(g)c(g)c(g)c(g)c(g) in the estimates above. In the context of Mordell's Conjecture, Bombieri observed that the number of large points is bounded by a multiple of 7rkJac(C)(F)7rkJac(C)(F)7^(rkJac)(C)(F)7^{\mathrm{rk} J a c}(C)(F)7rkJac(C)(F).
Question 1.17. Can the base 7 in the estimate for the number of large points as in Theorem 1.6 be replaced by a function in ggggg that tends to 1 for g→∞g→∞g rarr oog \rightarrow \inftyg→∞ ?
Alpoge [2] used the Kabatiansky-Levenshtein estimates on spherical codes to improve on the constant 7 in genus 2. It is quite possible that Alpoge's approach will shed light on this last question.
Concerning the constant c(g)c(g)c(g)c(g)c(g) in Theorem 1.16, we pose the following two questions which also cover the moderate, i.e., nonlarge, points. They were inspired by questions of Helfgott.
Question 1.18. Can we choose the c(g)c(g)c(g)c(g)c(g) in Theorem 1.16 such that there exists B≥1B≥1B >= 1B \geq 1B≥1 with c(g)≤Bc(g)≤Bc(g) <= Bc(g) \leq Bc(g)≤B for all integers g≥2g≥2g >= 2g \geq 2g≥2 ?
Question 1.19. Can we choose the c(g)c(g)c(g)c(g)c(g) in Theorem 1.16 with limg→∞c(g)=1limg→∞ c(g)=1lim_(g rarr oo)c(g)=1\lim _{g \rightarrow \infty} c(g)=1limg→∞c(g)=1 ?
Recently, Gao, Ge, and Kühne [32] completed the proof of the Uniform MordellLang Conjecture for a subvariety VVVVV of a polarized abelian variety AAAAA of any dimension. Uniformity here amounts to bounding the number of irreducible components of the Zariski closure in Theorem 1.2 from above by c′(dimA,degV)c(dimA,degV)rc′(dimâ¡A,degâ¡V)c(dimâ¡A,degâ¡V)rc^(')(dim A,deg V)c(dim A,deg V)^(r)c^{\prime}(\operatorname{dim} A, \operatorname{deg} V) c(\operatorname{dim} A, \operatorname{deg} V)^{r}c′(dimâ¡A,degâ¡V)c(dimâ¡A,degâ¡V)r. Their result holds over all base fields in characteristic 0 .
We refer to the comprehensive survey by Gao [31] that gives an overview of these recent developments and how they are interlinked.
We review here briefly the main properties of the Weil height. For a thorough treatment, we refer to [9, CHAPTERS 1 AND 2] or [37, PART B].
We begin by defining the height of a rational point on projective space PnPnP^(n)\mathbb{P}^{n}Pn.
Definition 2.1. Let P∈Pn(Q)P∈Pn(Q)P inP^(n)(Q)P \in \mathbb{P}^{n}(\mathbb{Q})P∈Pn(Q). There exist projective coordinates (x0,…,xn)∈x0,…,xn∈(x_(0),dots,x_(n))in\left(x_{0}, \ldots, x_{n}\right) \in(x0,…,xn)∈Zn+1∖{0}Zn+1∖{0}Z^(n+1)\\{0}\mathbb{Z}^{n+1} \backslash\{0\}Zn+1∖{0} of P=[x0:⋯:xn]P=x0:⋯:xnP=[x_(0):cdots:x_(n)]P=\left[x_{0}: \cdots: x_{n}\right]P=[x0:⋯:xn] with gcd(x0,…,xn)=1gcdâ¡x0,…,xn=1gcd(x_(0),dots,x_(n))=1\operatorname{gcd}\left(x_{0}, \ldots, x_{n}\right)=1gcdâ¡(x0,…,xn)=1. Then we set
The vector (x0,…,xn)x0,…,xn(x_(0),dots,x_(n))\left(x_{0}, \ldots, x_{n}\right)(x0,…,xn) is uniquely determined up to a sign, and so h(P)h(P)h(P)h(P)h(P) is well defined. For example, h([2:4:6])=h([1:2:3])=h([1/3:2/3:1])=log3h([2:4:6])=h([1:2:3])=h([1/3:2/3:1])=logâ¡3h([2:4:6])=h([1:2:3])=h([1//3:2//3:1])=log 3h([2: 4: 6])=h([1: 2: 3])=h([1 / 3: 2 / 3: 1])=\log 3h([2:4:6])=h([1:2:3])=h([1/3:2/3:1])=logâ¡3.
The following theorem is a straightforward consequence of the definition of the Weil height.
Theorem 2.2 (Northcott property). The set {P∈Pn(Q):h(P)≤B}P∈Pn(Q):h(P)≤B{P inP^(n)(Q):h(P) <= B}\left\{P \in \mathbb{P}^{n}(\mathbb{Q}): h(P) \leq B\right\}{P∈Pn(Q):h(P)≤B} is finite for all BBBBB.
Defining the height of an algebraic point in Pn(Q¯)Pn(Q¯)P^(n)( bar(Q))\mathbb{P}^{n}(\overline{\mathbb{Q}})Pn(Q¯) requires some basic algebraic number theory. Indeed, let KKKKK be a number field. We let MKMKM_(K)M_{K}MK denote the set of absolute values |⋅|:K→[0,∞)|â‹…|:K→[0,∞)|*|:K rarr[0,oo)|\cdot|: K \rightarrow[0, \infty)|â‹…|:K→[0,∞) that extend either the standard absolute value on QQQ\mathbb{Q}Q or a ppppp-adic absolute value for some prime ppppp. Then MKMKM_(K)M_{K}MK is called the set of places of KKKKK. For each v∈MKv∈MKv inM_(K)v \in M_{K}v∈MK, one sets dv=[Kv:Qw]dv=Kv:Qwd_(v)=[K_(v):Q_(w)]d_{v}=\left[K_{v}: \mathbb{Q}_{w}\right]dv=[Kv:Qw] where KvKvK_(v)K_{v}Kv is a completion of KKKKK with respect to vvvvv and QwQwQ_(w)\mathbb{Q}_{w}Qw is the completion of QQQ\mathbb{Q}Q in KvKvK_(v)K_{v}Kv with respect to w=v|Qw=vQw=v|_(Q)w=\left.v\right|_{\mathbb{Q}}w=v|Q.
Definition 2.3. Let P∈Pn(Q¯)P∈Pn(Q¯)P inP^(n)( bar(Q))P \in \mathbb{P}^{n}(\overline{\mathbb{Q}})P∈Pn(Q¯) and let KKKKK be a number field such that P=[x0:⋯:xn]P=x0:⋯:xnP=[x_(0):cdots:x_(n)]P=\left[x_{0}: \cdots: x_{n}\right]P=[x0:⋯:xn] where (x0,…,xn)∈Kn+1∖{0}x0,…,xn∈Kn+1∖{0}(x_(0),dots,x_(n))inK^(n+1)\\{0}\left(x_{0}, \ldots, x_{n}\right) \in K^{n+1} \backslash\{0\}(x0,…,xn)∈Kn+1∖{0}. The absolute logarithmic Weil height, or just Weil height, is
holds for all x∈K∖{0}x∈K∖{0}x in K\\{0}x \in K \backslash\{0\}x∈K∖{0}. This guarantees that the right-hand side of (2.1) is independent of the choice of projective coordinates of PPPPP. In particular, we may assume that some projective coordinate of PPPPP equals 1. Thus h(P)≥0h(P)≥0h(P) >= 0h(P) \geq 0h(P)≥0 for all P∈Pn(Q¯)P∈Pn(Q¯)P inP^(n)( bar(Q))P \in \mathbb{P}^{n}(\overline{\mathbb{Q}})P∈Pn(Q¯). Moreover, h(P)h(P)h(P)h(P)h(P) is independent of the field KKKKK containing the projective coordinates.
For applications to diophantine geometry, it is useful to have a height function defined on algebraic points of an irreducible projective variety VVVVV defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯. But without additional data there is no reasonable way to define a height on V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯).
However, if VVVVV is a subvariety of the projective space PnPnP^(n)\mathbb{P}^{n}Pn, then we may restrict the Weil height h:Pn(Q¯)→Rh:Pn(Q¯)→Rh:P^(n)( bar(Q))rarrRh: \mathbb{P}^{n}(\overline{\mathbb{Q}}) \rightarrow \mathbb{R}h:Pn(Q¯)→R to a function V(Q¯)→RV(Q¯)→RV( bar(Q))rarrRV(\overline{\mathbb{Q}}) \rightarrow \mathbb{R}V(Q¯)→R. Slightly more generally, if V→PnV→PnV rarrP^(n)V \rightarrow \mathbb{P}^{n}V→Pn is an immersion, then we may pull back the Weil height to V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯).
Recall that an immersion V→PnV→PnV rarrP^(n)V \rightarrow \mathbb{P}^{n}V→Pn is induced by a tuple of (n+1)(n+1)(n+1)(n+1)(n+1) global sections of a very ample invertible sheaf on VVVVV. Conversely, given a very ample invertible sheaf LLL\mathscr{L}L on VVVVV, we can fix a basis of the vector space of global sections of LLL\mathscr{L}L and obtain an immersion L:V→PnL:V→PnL:V rarrP^(n)\mathscr{L}: V \rightarrow \mathbb{P}^{n}L:V→Pn. So we obtain a function h∘L:V(Q¯)→[0,∞)h∘L:V(Q¯)→[0,∞)h@L:V( bar(Q))rarr[0,oo)h \circ \mathscr{L}: V(\overline{\mathbb{Q}}) \rightarrow[0, \infty)h∘L:V(Q¯)→[0,∞). There is a wrinkle here, this function depends not only on (V,L)(V,L)(V,L)(V, \mathscr{L})(V,L) but also on the basis of the vector space of global sections. A different basis will lead to a function V(Q¯)→[0,∞)V(Q¯)→[0,∞)V( bar(Q))rarr[0,oo)V(\overline{\mathbb{Q}}) \rightarrow[0, \infty)V(Q¯)→[0,∞) that differs from h∘Lh∘Lh@Lh \circ \mathscr{L}h∘L by a bounded function on V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯). We define hV,LhV,Lh_(V,L)h_{V, \mathscr{L}}hV,L to be the equivalence class of functions V(Q¯)→RV(Q¯)→RV( bar(Q))rarrRV(\overline{\mathbb{Q}}) \rightarrow \mathbb{R}V(Q¯)→R modulo bounded functions that contains h∘ιLh∘ιLh@iotaLh \circ \iota \mathscr{L}h∘ιL.
If LLL\mathscr{L}L is an ample invertible sheaf on VVVVV, then there exists an integer n≥1n≥1n >= 1n \geq 1n≥1 such that L⊗nL⊗nL^(ox n)\mathscr{L}^{\otimes n}L⊗n is very ample. We then define hV,L=1nhV,L⊗nhV,L=1nhV,L⊗nh_(V,L)=(1)/(n)h_(V,Lox n)h_{V, \mathscr{L}}=\frac{1}{n} h_{V, \mathscr{L} \otimes n}hV,L=1nhV,L⊗n; this is again only defined up to a bounded function on V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯). The equivalence class does not depend on the choice of nnnnn.
Finally, an arbitrary invertible sheaf LLL\mathscr{L}L in the Picard group Pic(V)Picâ¡(V)Pic(V)\operatorname{Pic}(V)Picâ¡(V) of VVVVV is of the form F⊗M⊗(−1)F⊗M⊗(−1)FoxM^(ox(-1))\mathscr{F} \otimes \mathcal{M}^{\otimes(-1)}F⊗M⊗(−1) with FFF\mathcal{F}F and MMM\mathcal{M}M ample on VVVVV. The difference hV,F−hV,MhV,F−hV,Mh_(V,F)-h_(V,M)h_{V, \mathcal{F}}-h_{V, \mathcal{M}}hV,F−hV,M is well defined up to a bounded function on V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯). It does not depend on the pair F,MF,MF,M\mathcal{F}, \mathcal{M}F,M with difference LLL\mathscr{L}L, and we denote it by hV,LhV,Lh_(V,L)h_{V, \mathscr{L}}hV,L. It is called the Weil height attached to (V,L)(V,L)(V,L)(V, \mathscr{L})(V,L).
Theorem 2.4. Let us keep the notation above. In particular, VVVVV is an irreducible projective variety defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯.
(i) The association L↦hV,LL↦hV,LL|->h_(V,L)\mathscr{L} \mapsto h_{V, \mathscr{L}}L↦hV,L is a group homomorphism with target the group of real-valued maps V(Q¯)→RV(Q¯)→RV( bar(Q))rarrRV(\overline{\mathbb{Q}}) \rightarrow \mathbb{R}V(Q¯)→R modulo bounded functions.
(ii) For VVVVV equal to projective space and LLL\mathscr{L}L the hyperplane bundle O(1)O(1)O(1)\mathcal{O}(1)O(1), the Weil height from Definition 2.3 represents hPn,O(1)hPn,O(1)h_(P^(n),O(1))h_{\mathbb{P}^{n}, \mathcal{O}(1)}hPn,O(1).
(iii) Suppose WWWWW is a further irreducible projective variety defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ and fffff : W→VW→VW rarr VW \rightarrow VW→V is a morphism. For all L∈Pic(V)L∈Picâ¡(V)Lin Pic(V)\mathscr{L} \in \operatorname{Pic}(V)L∈Picâ¡(V) we have hV,L∘f=hW,f∗LhV,L∘f=hW,f∗Lh_(V,L)@f=h_(W,f)**Lh_{V, \mathscr{L}} \circ f=h_{W, f} * \mathscr{L}hV,L∘f=hW,f∗L. As usual, this equality is understood as an equality of equivalence classes of functions.
(iv) Suppose L∈Pic(V)L∈Picâ¡(V)Lin Pic(V)\mathscr{L} \in \operatorname{Pic}(V)L∈Picâ¡(V) admits a nonzero global section sssss. Then hV,LhV,Lh_(V,L)h_{V, \mathscr{L}}hV,L is bounded from below on the complement of the vanishing locus of sssss. In particular, hV,LhV,Lh_(V,L)h_{V, \mathscr{L}}hV,L is bounded from below on a Zariski open and dense subset of VVVVV.
Suppose that VVVVV is defined over a number field F⊆Q¯F⊆Q¯F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯ and L∈Pic(V)L∈Picâ¡(V)Lin Pic(V)\mathscr{L} \in \operatorname{Pic}(V)L∈Picâ¡(V) is ample. Then the Northcott property holds for points of bounded degree, i.e.,
{P∈V(F¯):hV,L′(P)≤B and [F(P):F]≤D}P∈V(F¯):hV,L′(P)≤B and [F(P):F]≤D{P in V(( bar(F))):h_(V,L)^(')(P) <= B" and "[F(P):F] <= D}\left\{P \in V(\bar{F}): h_{V, \mathscr{L}}^{\prime}(P) \leq B \text { and }[F(P): F] \leq D\right\}{P∈V(F¯):hV,L′(P)≤B and [F(P):F]≤D}
is finite where hV,L′hV,L′h_(V,L)^(')h_{V, \mathscr{L}}^{\prime}hV,L′ denotes any representative of hV,LhV,Lh_(V,L)h_{V, \mathscr{L}}hV,L.
Let VVVVV be an irreducible projective variety defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯. We conclude this section by discussing a powerful tool to translate geometric information, here on intersection numbers, into an inequality of heights. The basic question is the following. Given invertible sheaves FFF\mathcal{F}F and MMM\mathcal{M}M on VVVVV, under what conditions can one bound hV,MhV,Mh_(V,M)h_{V, \mathcal{M}}hV,M from above in terms of hV,FhV,Fh_(V,F)h_{V, \mathcal{F}}hV,F ?
(i) We first consider the special case V=PnV=PnV=P^(n)V=\mathbb{P}^{n}V=Pn. As Pic(Pn)Picâ¡PnPic(P^(n))\operatorname{Pic}\left(\mathbb{P}^{n}\right)Picâ¡(Pn) is isomorphic to ZZZ\mathbb{Z}Z, any Weil height is some integral multiple of hPn,O(1)hPn,O(1)h_(P^(n),O(1))h_{\mathbb{P}^{n}, \mathcal{O}(1)}hPn,O(1). So hV,FhV,Fh_(V,F)h_{V, \mathcal{F}}hV,F and hV,MhV,Mh_(V,M)h_{V, \mathcal{M}}hV,M are ZZZ\mathbb{Z}Z-linearly dependent.
(ii) Let us again suppose that VVVVV is general and that FFF\mathscr{F}F is ample. Then there exists an integer k≥1k≥1k >= 1k \geq 1k≥1 such that F⊗k⊗M⊗(−1)F⊗k⊗M⊗(−1)Fox k oxM^(ox(-1))\mathcal{F} \otimes k \otimes \mathcal{M}^{\otimes(-1)}F⊗k⊗M⊗(−1) is ample. So for some positive integer l≥1l≥1l >= 1l \geq 1l≥1 the power F⊗kl⊗M⊗(−l)F⊗kl⊗M⊗(−l)F^(ox kl)oxM^(ox(-l))\mathscr{F}^{\otimes k l} \otimes \mathcal{M}^{\otimes(-l)}F⊗kl⊗M⊗(−l) is very ample. In particular, it admits a global section that does not vanish at a prescribed point of V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯). Theorem 2.4, parts (i) and (iv), imply
klhV,F−lhV,M=hV,F⊗kl⊗M⊗(−l)≥0klhV,F−lhV,M=hV,F⊗kl⊗M⊗(−l)≥0klh_(V,F)-lh_(V,M)=h_(V,Fox kl oxMox(-l)) >= 0k l h_{V, \mathcal{F}}-l h_{V, \mathcal{M}}=h_{V, \mathcal{F} \otimes k l \otimes \mathcal{M} \otimes(-l)} \geq 0klhV,F−lhV,M=hV,F⊗kl⊗M⊗(−l)≥0
this must be parsed as an inequality between functions on V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯) defined up to addition of a bounded function. We conclude
(iii) For some applications, such as Theorem 1.12, the ampleness hypothesis on FFF\mathcal{F}F in (i) is not flexible enough. Moreover, we would like some way to estimate the factor kkkkk in (2.2) from above. We now describe a criterion of Siu that provides a solution to these two issues.
An invertible sheaf L∈Pic(V)L∈Picâ¡(V)Lin Pic(V)\mathscr{L} \in \operatorname{Pic}(V)L∈Picâ¡(V) is called big if
lim infk→∞dimH0(V,L⊗k)kdimV>0lim infk→∞ dimâ¡H0V,L⊗kkdimâ¡V>0l i m   i n f_(k rarr oo)(dim H^(0)(V,L^(ox k)))/(k^(dim V)) > 0\liminf _{k \rightarrow \infty} \frac{\operatorname{dim} H^{0}\left(V, \mathscr{L}^{\otimes k}\right)}{k^{\operatorname{dim} V}}>0lim infk→∞dimâ¡H0(V,L⊗k)kdimâ¡V>0
here H0(V,L)H0(V,L)H^(0)(V,L)H^{0}(V, \mathscr{L})H0(V,L) denotes the vector space of global sections of LLL\mathscr{L}L.
If LLL\mathscr{L}L is a big invertible sheaf, then L⊗kL⊗kL^(ox k)\mathscr{L}^{\otimes k}L⊗k has a nonzero global section for some k≥1k≥1k >= 1k \geq 1k≥1. Then using (i) and (iv) of Theorem 2.4 we see that hV,L=1khV,L⊗khV,L=1khV,L⊗kh_(V,L)=(1)/(k)h_(V,Lox k)h_{V, \mathscr{L}}=\frac{1}{k} h_{V, \mathscr{L} \otimes k}hV,L=1khV,L⊗k is bounded from below on a Zariski open and dense subset of VVVVV.
For example, if L=F⊗M⊗(−1)L=F⊗M⊗(−1)L=FoxM^(ox(-1))\mathscr{L}=\mathscr{F} \otimes \mathcal{M}^{\otimes(-1)}L=F⊗M⊗(−1) is big, then, again by Theorem 2.4(i), we find hV,F≥hV,MhV,F≥hV,Mh_(V,F) >= h_(V,M)h_{V, \mathcal{F}} \geq h_{V, \mathcal{M}}hV,F≥hV,M on a Zariski open and dense subset of VVVVV.
We now come Siu's Criterion; it ensures that F⊗M⊗(−1)F⊗M⊗(−1)FoxM^(ox(-1))\mathscr{F} \otimes \mathcal{M}^{\otimes(-1)}F⊗M⊗(−1) is big. An invertible sheaf L∈Pic(V)L∈Picâ¡(V)Lin Pic(V)\mathscr{L} \in \operatorname{Pic}(V)L∈Picâ¡(V) is called nef, or numerically effective, if (L⋅[C])≥0(Lâ‹…[C])≥0(L*[C]) >= 0(\mathscr{L} \cdot[C]) \geq 0(Lâ‹…[C])≥0 for all irreducible curves C⊆VC⊆VC sube VC \subseteq VC⊆V.
Siu's Criterion requires that FFF\mathscr{F}F and MMM\mathcal{M}M are both nef and that the intersection numbers on VVVVV satisfy (F⋅dimV)>(dimV)(F⋅(dimV−1)⋅M)(Fâ‹…dimâ¡V)>(dimâ¡V)(Fâ‹…(dimâ¡V−1)â‹…M)(F*dim V) > (dim V)(F*(dim V-1)*M)(\mathcal{F} \cdot \operatorname{dim} V)>(\operatorname{dim} V)(\mathcal{F} \cdot(\operatorname{dim} V-1) \cdot \mathcal{M})(Fâ‹…dimâ¡V)>(dimâ¡V)(Fâ‹…(dimâ¡V−1)â‹…M). With these hypotheses F⊗M⊗(−1)F⊗M⊗(−1)FoxM^(ox(-1))\mathcal{F} \otimes \mathcal{M}^{\otimes(-1)}F⊗M⊗(−1) is big; see [42, THEOREM 2.2.15].
Say FFF\mathscr{F}F and MMM\mathcal{M}M are nef and (F⋅dimV)>0(Fâ‹…dimâ¡V)>0(F*dim V) > 0(\mathscr{F} \cdot \operatorname{dim} V)>0(Fâ‹…dimâ¡V)>0. Let kkkkk and lllll be positive integers with
holds on some Zariski open and dense U⊆VU⊆VU sube VU \subseteq VU⊆V.
This allows us to compare the heights hV,MhV,Mh_(V,M)h_{V, \mathcal{M}}hV,M and hV,FhV,Fh_(V,F)h_{V, \mathcal{F}}hV,F if we have information on the intersection numbers, at least on a rather large subset of V(Q¯)V(Q¯)V( bar(Q))V(\overline{\mathbb{Q}})V(Q¯).
Yuan [65] proved an arithmetic version of this criterion in his work on equidistribution. The author [34] used Siu's Criterion to study unlikely intersections in abelian varieties.
2.2. The canonical height on an abelian variety
Let F⊆Q¯F⊆Q¯F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯ be a number field and AAAAA an abelian variety defined over FFFFF. If LLL\mathscr{L}